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Chuyên đề nhị thức newton (niu tơn) – lê văn đoàn

ĐỀ CƯƠNG HỌC TẬP LỚP 11

SỞ GD & ĐT TP. HỒ CHÍ MINH
TRUNG TÂM THẾ VINH – 45 LÊ THÚC HOẠCH – Q. TÂN PHÚ

Mơn: Tốn, Năm học: 2019 – 2020

§ 3. NHỊ THỨC NEWTON


1. Nhò thức Newton: Cho a, b là các số thực và n  . Ta có:
(a  b)n 

n

C
k 0

 (x  2)4 

4


C
k 0

k
4

k
n

.a n k .b k  C n0a n  C n1a n 1b  C n2a n2b 2      C nn 1ab n 1  C nnb n

.x 4k .2k  C 40 .x 4 .20  C 41.x 3 .21  C 42 .x 2 .22  C 43 .x 1.23  C 44 .x 0 .24

 x 4  8x 3  24x 2  32x  16.

 (x  2y)5  ........................................................................................................................................
..................................................................................................................................................................
6


1
 x    ........................................................................................................................................
x 

..................................................................................................................................................................
6


1
 2x    ......................................................................................................................................
x 

..................................................................................................................................................................

2. Nhận xét:
 Trong khai triển (a  b)n có n  1 số hạng và các hệ số của các cặp số hạng cách đều số
hạng đầu và số hạng cuối thì bằng nhau: C nk  C nnk .
 Số hạng tổng qt dạng: Tn 1  C nk .a n k .b k và số hạng thứ N thì k  N  1.
 Trong khai triển (a  b)n thì dấu đan nhau, nghĩa là , rồi , rồi , ….…
 Số mũ của a giảm dần, số mũ của b tăng dần nhưng tổng số mũ a và b bằng n.

3. Tam giác Pascal:
Các hệ số của khai triển: (a  b)0, (a  b)1, (a  b)2, ..., (a  b )n có thể xếp thành một tam
giác gọi là tam giác PASCAL.

n 0:
n 1:

1
1 1

n 2:

1 2

1

n 3:

1

3

n 4:
n 5:

1 4 6 4 1
1 5 10 10 5

n 6:

1

6

15 20 15 6

n 7:

1

7

21 35 35 21 7

3

Hằng đẳng thức
PASCAL

C nk11  C nk1

1


1

C nk

1
1

................................................

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Trang - 127 -


ẹE CệễNG HOẽC TAP LễP 11

S GD & T TP. H CH MINH
TRUNG TM TH VINH 45 Lấ THC HOCH Q. TN PH

Mụn: Toỏn, Nm hc: 2019 2020

Dng toỏn 1. Tỡm h s hoc s hng trong khai trin nh thc Newton
n

k
n

Cn nh: Tk 1 C a

n k

.b

n

k

v x .x

m

x

m n

1. Tỡm s hng khụng cha x (c lp vi x )


1
trong khai trin x , x 0.

x


1
trong khai trin x 3 2 , x 0.

x

1
v n 5.
x2

..............................................................................
k

1
S hng tng quỏt: Tk 1 C 5k .(x 3 )5k . 2
x

..............................................................................
..............................................................................
..............................................................................

k

(1)
C 5k .(1)k .x 155k .
2k
x
S hng khụng cha x 15 5k 0 k 3.
C 5k .x 153k .

2. Tỡm s hng khụng cha x (c lp vi x )
12

5

Ta cú: a x 3, b

x
xn
xn
, m x n m , (x .y )n x n .y n , n
y
x
y

Vy s hng cn tỡm l C 53 .(1)3 10.
3. Tỡm s hng khụng cha x (c lp vi x )
10


1
trong khai trin x 4 , x 0.

x

..............................................................................
..............................................................................
S: 924. ..............................................................
4. Tỡm s hng khụng cha x (c lp vi x )
12

x
3
trong khai trin , x 0.
3 x

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S: 45. .................................................................

S: 924. ..............................................................

5. Tỡm s hng khụng cha x (c lp vi x )

6. Tỡm s hng khụng cha x (c lp vi x )

6


1
trong khai trin 2x 2 , x 0.

x

10


1
trong khai trin 2x , x 0.

x

...............................................................................

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S: 240. ...............................................................

S: 8064. ........................................................

Biên soạn và giảng dạy: Ths. Lê Văn Đoàn 0933.755.607 0929.031.789

Trang - 128 -


ẹE CệễNG HOẽC TAP LễP 11

S GD & T TP. H CH MINH
TRUNG TM TH VINH 45 Lấ THC HOCH Q. TN PH

Mụn: Toỏn, Nm hc: 2019 2020

7. Tỡm h s ca s hng cha x 16 trong khai 8. Tỡm h s ca s hng cha x 6 trong khai
trin nh thc (x 2 2x )10 .

trin nh thc (1 3x )11.

Ta cú: a x 2, b 2x, n 10.

..............................................................................

k
S hng TQ: Tk 1 C 10
.(x 2 )10k .(2x )k

..............................................................................

C 10k .x 2(10k ).(2)k .x k

..............................................................................

C 10k .(2)k .x 202k .x k

..............................................................................

C 10k .(2)k x 20k .

..............................................................................

Vỡ cú s hng x 16 20 k 16 k 4.

..............................................................................

H s cn tỡm l C 104 .(2)4 3360.

S: 336798. .......................................................

9. Tỡm h s ca s hng cha x 15 trong khai 10. Tỡm h s ca s hng cha x 4 trong khai
trin nh thc (3x x 2 )12 .

trin nh thc (x 3)9 .

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S: 4330260. ...................................................

S: 30618. ......................................................

11. Tỡm h s ca s hng cha x 12y 13 trong 12. Tỡm h s ca s hng cha x 8y 9 trong
khai trin nh thc (x y )25 .

khai trin nh thc (2x 3y )17 .

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S: 5200300. ......................................................

S: C 179 .2179.39 122494394880. ...................

13. Tỡm h s ca s hng cha x 6y 7 trong 14. Tỡm h s ca s hng cha x 25y 10 trong
khai trin nh thc (2x y )13 .

khai trin nh thc (x 3 xy )15 .

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S: 109824. ........................................................

S: 3003. ............................................................

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Trang - 129 -


ẹE CệễNG HOẽC TAP LễP 11

S GD & T TP. H CH MINH
TRUNG TM TH VINH 45 Lấ THC HOCH Q. TN PH

15.



Tỡm h s ca s hng cha x 4 trong 16.

khai trin nh thc (1 x 3x 2 )10.
Ta cú: (1 x 3x 2 )10 [1 (x 3x 2 )]10

(a 1, b x 3x 2, n 10)


10

C 10k .110k.(x 3x 2 )k
k 0

10

C
k 0

k
10

(x 3x 2 )k

2

(a x, b 3x , n k )


10

k

C .C
k 0
10

k
10

p 0

p
k

.x k p .(3x 2 )p

k

C 10k .C kp .x k p .3p.x 2 p
k 0 p 0



10

k

C
k 0 p 0

k
10

.C kp .3p x k p .



Mụn: Toỏn, Nm hc: 2019 2020

Tỡm h s ca s hng cha x 17 trong

khai trin nh thc (1 x 2x 2 )10 .
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4

Vỡ cú s hng x k p 4 vi iu kin
0 p k 10, (p, k ) nờn cú bng:

..............................................................................

p
0 1 2
3
k 4 p 4 3 2 1 (L)

..............................................................................

Do ú h s ca s hng cha x 4 l:

..............................................................................

C 104 C 40 .30 C 103 C 31 .31 C 102 C 22 .32 1695.

S: 38400. .........................................................

17.



..............................................................................

Tỡm h s ca s hng cha x 8 trong 18.

khai trin nh thc (1 x 2 x 3 )8 .



Tỡm h s ca s hng cha x 3 trong

khai trin nh thc (x 2 x 1)5 .

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S: 238. ...............................................................

S: 5. ..................................................................

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Trang - 130 -


ẹE CệễNG HOẽC TAP LễP 11

S GD & T TP. H CH MINH
TRUNG TM TH VINH 45 Lấ THC HOCH Q. TN PH

19.



Tỡm h s ca s hng cha x 10 trong 20.

khai trin P(x ) (1 x x 2 x 3 )5 .



Mụn: Toỏn, Nm hc: 2019 2020

Tỡm h s ca s hng cha x 5 trong

khai trin P(x ) (1 x x 2 x 3 )10 .

Ta cú: P(x ) [(1 x ) x 2 (1 x )]5

..............................................................................

[(1 x )(1 x 2 )]5

..............................................................................

(1 x )5 .(1 x 2 )5

..............................................................................




5

5

..............................................................................

k 0

p 0

..............................................................................

.x k 2 p .

..............................................................................

C 5k .15k.x k .C 5p .15p.x 2p
5

5

C C
k
5

k 0 p 0

p
5

..............................................................................

Vỡ cú s hng x

10

nờn k 2p 10 v cú bng

p
2
3
4
5
6
k 10 2p
6
4
2
0 2
(0 p; k 5) (L) (N) (N) (N) (L)
Do ú h s ca s hng cha x 10 l
4
5

3
5

2
5

4
5

0
5

5
5

C C C C C C 101.

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..............................................................................
S: 1902. ............................................................

21. Xột P(x ) x (1 2x )5 x 2 (1 3x )10 . Tỡm 22. Xột P(x ) x (2x 1)6 (3x 1)8 . Tỡm h
h s x 5 trong khai trin P (x ).
5

10

..............................................................................

k 0

p 0

..............................................................................

Ta cú P (x ) x C 5k (2x )k x 2 C 10p (3x )p

.x .(2)k .x k C 10p .x 2 .3p.x p

10

..............................................................................

p 0

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S: 3320. ............................................................

S: 13368. ......................................................



5

s x 5 trong khai trin P (x ).

C
k 0

k
5

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Trang - 131 -


ẹE CệễNG HOẽC TAP LễP 11

S GD & T TP. H CH MINH
TRUNG TM TH VINH 45 Lấ THC HOCH Q. TN PH

23.



Tỡm h s ca x 6 trong khai trin biu 24.
4

4

4x 2 4x 1

Ta cú: P (x ) (2x 1)

4


Tỡm h s ca s hng cha x 10 trong
2


1
thc P (x ) (2x 1) x 2 x .

4

x 2

khai trin P (x ) x 1 (x 2)15 .
4


6

6



Mụn: Toỏn, Nm hc: 2019 2020

..............................................................................
..............................................................................

4

(2x 1)2

(2x 1)6 .

4


8
(2x 1)
(2x 1)6 .
44
1

(2x 1)14 .
256
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ỏp s:
25.



3003
. .....................................................
4

Tỡm h s ca x 5 trong khai trin sau:

S: a10
26.



1 9 9
.C .2 2956096. .....................
16 19

Tỡm h s ca x 5 trong khai trin sau:

(2x 1)4 (2x 1)5 (2x 1)6 (2x 1)7 .

(x 1)6 (x 1)7 (x 1)8 (x 1)12

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ỏp s: 896. ........................................................

S: 1715. ...........................................................

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Trang - 132 -


ẹE CệễNG HOẽC TAP LễP 11

S GD & T TP. H CH MINH
TRUNG TM TH VINH 45 Lấ THC HOCH Q. TN PH

27. Gi s cú khai trin a thc:

Mụn: Toỏn, Nm hc: 2019 2020

28. Gi s cú khai trin a thc:

(1 2x )n a 0 a1x a 2x 2 a n x n .

(1 4x )n a 0 a1x a 2x 2 a n x n .

Tỡm a 5 , bit rng a 0 a1 a2 71.

Tỡm a5 bit a 0 a1 a 2 1197.

Cú (1 2x )n

n

C
k 0

k
n

(2x )k

n

C
k 0

k
n

(2)k x k

..............................................................................
k
n

k

Dng tng quỏt ca h s l ak C (2) .
1
n

..............................................................................

..............................................................................

0

Vi k 0 a 0 C (2) 1.

..............................................................................

1
n

Vi k 1 a1 2C 2n.

..............................................................................

2
n

Vi k 2 a 2 4C .

..............................................................................

Theo bi ta cú: a 0 a1 a 2 71

..............................................................................

2
n

1 2n 4C 71 .................................

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ỏp s: a 5 C 75 (2)5 672. .......................

S: a5 1317888. .........................................

29.



Gi s cú khai trin a thc:
n

30.

Gi s cú khai trin a thc:

(1 2x ) a 0 a1x a 2x an x .

(1 2x )n ao a1x a 2x 2 a n x n .

Tỡm n bit a 0 8a1 2a 2 1.

Bit a 3 2014a 2 . Tỡm n.

2

n



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S: n 5. ..........................................................

S: n 6044. ....................................................

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Trang - 133 -


S GD & T TP. H CH MINH
TRUNG TM TH VINH 45 Lấ THC HOCH Q. TN PH

ẹE CệễNG HOẽC TAP LễP 11
Mụn: Toỏn, Nm hc: 2019 2020

BI TP V NH 1
Cõu 1. Cú bao nhiờu s hng trong khai trin nh thc (2x 3)2018 .
A. 2017.

B. 2018.

..............................................................................................

C. 2019.

D. 2020.

..............................................................................................

Cõu 2. Trong khai trin (a b)n , s hng tng quỏt ca khai trin l
A. C nka n kb k .

B. C nk 1a n 1b n k 1 .

..............................................................................................

C. C nka n kb n k .

D. C nk 1a n k 1b k 1 .

..............................................................................................

Cõu 3. Tỡm s hng cha x 3y 3 trong khai trin (x 2y )6 thnh a thc.
A. 120x 3y 3 .

B. 160x 3y 3 .

..............................................................................................

C. 20x 3y 3 .

D. 8x 3y 3 .

..............................................................................................

Cõu 4. Tỡm h s ca s hng cha x 3 trong khai trin nh thc Niutn (2x 1)6 .
A. 160.

B. 960.

..............................................................................................

C. 960.

D. 160.

..............................................................................................

Cõu 5. Gi s cú khai trin (1 2x )7 a 0 a1x a 2x 2 a 7 x 7 . Tỡm a 5 .
A. 672x 5 .

B. 672.

..............................................................................................

C. 672x 5 .

D. 672.

..............................................................................................

Cõu 6. Tỡm h s ca x 6 trong khai trin thnh a thc ca (2 3x )10.
A. C 106 .26.(3)4 .

B. C 106 .24.(3)6 .

..............................................................................................

C. C 104 .26.(3)4 . D. C 106 .24.36.

..............................................................................................
10


1
Cõu 7. S hng khụng cha x trong khai trin P (x ) 2x l s hng th

x
A. 6.

B. 7.

..............................................................................................

C. 8.

D. 9.

..............................................................................................
12

3 x
Cõu 8. H s ca s hng cha x trong khai trin nh thc (vi x 0 ) l
x 3
6

220

729
220 6
x .
C.
729
A.

220 6
x .
729
220

D.
729
B.

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Trang - 134 -


ẹE CệễNG HOẽC TAP LễP 11

S GD & T TP. H CH MINH
TRUNG TM TH VINH 45 Lấ THC HOCH Q. TN PH

Mụn: Toỏn, Nm hc: 2019 2020
6


1
Cõu 9. S hng khụng cha x trong khai trin 2x 2 l

x
A. 60.

B. 120.

..............................................................................................

C. 480.

D. 240.

..............................................................................................
9

1

Cõu 10. H s ca s hng cha x trong khai trin x 3 , (vi x 0 ) bng
x

3

A. 36.

B. 84.

C. 126.

D. 54.

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Cõu 11. S hng cha x 4 trong khai trin (2 x )7 thnh a thc l
A. 8C 74 .

B. C 74 .

..............................................................................................

C. 8C 74x 4 .

D. C 74x 4 .

..............................................................................................
45


1
Cõu 12. S hng khụng cha x trong khai trin x 2 l

x
5
A. C 45
.

5
B. C 45
.

15
C. C 45
.

15
D. C 45
.

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Cõu 13. Trong khai trin ca (1 3x )9 s hng th 3 theo s m tng dn ca x l
A. 180x 2 .

B. 120x 2 .

..............................................................................................

C. 324x 2 .

D. 4x 2 .

..............................................................................................
21


2
Cõu 14. Tỡm s hng khụng cha x trong khai trin nh thc Newton x 2

x
7
A. 27C 21
.

B. 28C 218 .

..............................................................................................

C. 28C 218 .

7
D. 27C 21
.

..............................................................................................
12


1
Cõu 15. Cho x l s thc dng, khai trin nh thc x 2 ta cú h s ca s hng cha

x

x m bng 495. Tp hp giỏ tr ca m l
A. {4; 8}.

B. {0}.

..............................................................................................

C. {0;12}.

D. {8}.

..............................................................................................

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Trang - 135 -


ẹE CệễNG HOẽC TAP LễP 11

S GD & T TP. H CH MINH
TRUNG TM TH VINH 45 Lấ THC HOCH Q. TN PH

Mụn: Toỏn, Nm hc: 2019 2020
n


1
Cõu 16. Bit h s ca s hng cha x trong khai trin 3x 2 l 34C n5 . Khi ú giỏ tr

x
ca n bng
3

A. 15.

B. 9.

C. 16.

D. 12.


Cõu 17.

B. 70.

C. 56.

D. 56.


B. 1272.

C. 1752.

D. 1272.


B. 263.

C. 632.

D. 956.


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..............................................................................................
..............................................................................................

H s ca x 4 trong khai trin a thc f (x ) x (1 x )5 x 2 (1 2x )10 bng

A. 965.

Cõu 20.

..............................................................................................

Tỡm h s x 5 trong khai trin x (2x 1)6 (x 3)8 .

A. 1752.

Cõu 19.

..............................................................................................

Tỡm h s ca s hng cha x 6 trong khai trin x 3 (1 x )8 .

A. 28.

Cõu 18.

..............................................................................................

..............................................................................................
..............................................................................................
..............................................................................................

Gi s (1 x x 2 )n a 0 a1x a 2x 2 a 2n x 2n . t S a 0 a2 a 2n ,

khi ú S bng
A.

3n 1

2

3n

2

..............................................................................................

D. 2n 1.

..............................................................................................

B.

3n 1
C.

2

..............................................................................................

..............................................................................................

P N BI TP V NH 1
1.C

2.A

3.B

4.D

5.B

6.B

7.A

8.A

9.A

10.B

11.C

12.C

13.C

14.D

15.C

16.B

17.C

18.D

19.A

20.A

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Trang - 136 -


ẹE CệễNG HOẽC TAP LễP 11

S GD & T TP. H CH MINH
TRUNG TM TH VINH 45 Lấ THC HOCH Q. TN PH

Mụn: Toỏn, Nm hc: 2019 2020

BI TP V NH 2
Cõu 1. S hng tng quỏt trong khai trin ca (1 2x )12 l
k
A. (1)k C 12
2x k .

k k k
B. C 12
2 x .

..............................................................................................

C. (1)k C 12k 2k x k . D. C 12k 2k x 12k .

..............................................................................................

Cõu 2. H s ca x 5 trong khai trin (1 x )12 l
A. 820.

B. 210.

..............................................................................................

C. 792.

D. 220.

..............................................................................................

Cõu 3. H s ca s hng cha x 3 trong khai trin (1 x )10 l
A. 30.

B. 120.

..............................................................................................

C. 120.

D. 30.

..............................................................................................

Cõu 4. H s ca x 10 trong biu thc P (2x 3x 2 )5 bng
A. 357.

B. 243.

..............................................................................................

C. 628.

D. 243.

..............................................................................................

Cõu 5. Trong khai trin biu thc (x y )21, h s ca s hng cha x 13y 8 l
A. 116280.

B. 293930.

..............................................................................................

C. 203490.

D. 1287.

..............................................................................................

Cõu 6. Trong khai trin (a 2b)8 , h s ca s hng cha a 4 .b 4 l
A. 560.

B. 70.

..............................................................................................

C. 1120.

D. 140.

..............................................................................................

Cõu 7. Tỡm h s ca x 6 trong khai trin thnh a thc ca (2 3x )10 .
A. C 106 .26.(3)4 .

B. C 106 .24.(3)6 .

..............................................................................................

C. C 104 .26.(3)4 D. C 106 .24.36.

..............................................................................................
10

1

Cõu 8. H s ca x trong khai trin x 3 bng
x

6

A. 792

B. 210

..............................................................................................

C. 165

D. 252

..............................................................................................
7


2
Cõu 9. Tỡm h s ca s hng cha x trong khai trin x 2 .

x
5

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Trang - 137 -


ẹE CệễNG HOẽC TAP LễP 11

S GD & T TP. H CH MINH
TRUNG TM TH VINH 45 Lấ THC HOCH Q. TN PH

Mụn: Toỏn, Nm hc: 2019 2020

A. 84.

B. 672.

..............................................................................................

C. 560.

D. 280.

..............................................................................................
6


1
Cõu 10. Tỡm s hng khụng cha x trong khai trin 2x 2 .

x
A. 15.

B. 240.

..............................................................................................

C. 240.

D. 15.

..............................................................................................
5

Cõu 11. Tỡm h s ca x

10


2
trong khai trin biu thc 3x 3 2 .

x

A. 240.

B. 810.

..............................................................................................

C. 810.

D. 240.

..............................................................................................
7

x 2 1
Cõu 12. Tỡm s hng cha x trong khai trin .
2
x
5

35 5
x .
16
16
C. x 5 .
35
A.

35 5
x .
16
16 5
D.
x .
35
B.

..............................................................................................
..............................................................................................
..............................................................................................

Cõu 13. Xột khai trin: (5x 1)2017 a2017x 2017 a 2016x 2016 a1x a 0 . Giỏ tr a 2000 bng
17
A. C 2017
.517.

17
B. C 2017
.517.

..............................................................................................

17
C. C 2017
.52000.

17
D. C 2017
.52000.

..............................................................................................
7


1
Cõu 14. H s ca x trong khai trin ca x 2 (2x 1)2 bng

x
2

A. 4.

B. 40.

..............................................................................................

C. 35.

D. 39.

..............................................................................................

Cõu 15. Tỡm h s ca x 5 trong khai trin P (x ) (x 1)6 (x 1)7 (x 1)12 .
A. 1715.

B. 1711.

..............................................................................................

C. 1287.

D. 1716.

..............................................................................................

Cõu 16.



Tỡm h s ca s hng cha x 9 trong khai trin nh thc Newton (1 2x )(3 x )11 .

A. 4620.

B. 1380.

..............................................................................................

C. 9405.

D. 2890.

..............................................................................................

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Trang - 138 -


ẹE CệễNG HOẽC TAP LễP 11

S GD & T TP. H CH MINH
TRUNG TM TH VINH 45 Lấ THC HOCH Q. TN PH

Cõu 17.



Mụn: Toỏn, Nm hc: 2019 2020

Tỡm h s ca x 5 trong khai trin P (x ) x (1 2x )5 x 2 (1 3x )10 .

A. 3240.

B. 3320.

C. 80.

D. 259200.

..............................................................................................
..............................................................................................
..............................................................................................
..............................................................................................

Cõu 18.



Cho khai trin (1 2x )9 a 0 a1x a 2x 2 a 9 x 9 . Tng a 0 a1 a 2 bng

A. 127.

B. 46.

C. 2816.

D. 163.

..............................................................................................
..............................................................................................
..............................................................................................
..............................................................................................

Cõu 19.



Tỡm h s ca x 7 trong khai trin f (x ) (1 3x 2x 3 )10 thnh a thc.

A. 204120.

B. 262440.

C. 4320.

D. 62640.

..............................................................................................
..............................................................................................
..............................................................................................
..............................................................................................

Cõu 20.



Tỡm h s ca s hng cha x 5 trong khai trin (1 x x 2 x 3 )10 .

A. 582.

B. 1902.

C. 7752.

D. 252.

..............................................................................................
..............................................................................................
..............................................................................................
..............................................................................................

P N BI TP V NH 2
1.C

2.C

3.B

4.D

5.C

6.C

7.B

8.B

9.D

10.B

11.C

12.C

13.C

14.D

15.A

16.C

17.B

18.A

19.D

20.B

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Trang - 139 -


S GD & T TP. H CH MINH
TRUNG TM TH VINH 45 Lấ THC HOCH Q. TN PH

ẹE CệễNG HOẽC TAP LễP 11
Mụn: Toỏn, Nm hc: 2019 2020

Dng toỏn 2. Chng minh hoc tớnh tng
n

(a b )n C nk .a n k .b k C n0a n C n1a n 1b C n2a n 2b 2 C nn 1ab n 1 C nnb n
k 0

S m ca a gim dn, s m ca b tng dn nhng tng s m a v b bng n.
Trong khai trin (a b)n thỡ du an nhau, ngha l , ri , ri , .
1
31. Chng minh: 316C 160 315C 16
314C 162 3C 1615 C 1616 216.

Suy lun:

S m ca s 3 gim dn Chn a 3.
Khụng cú s m ca s no tng Chn b 1 (vỡ 10 11 12 116 1).
Du an nhau (cng ri tr, cng tr..) nờn chn khai trin (a b)n (3 1)n .
Vỡ t hp dng: C 160 , C 161 , C 162 , ...C 1616 nờn chn n 16.
Li gii tham kho
Xột (3 1)16

16

C
k 0

16

16

k
16

.316k .(1)k 316C 160 315C 161 314C 162 3C 1615 C 1616

0
16

1
2 3 C 315C 16
314C 162 3C 1615 C 1616 (pcm).

32. Tớnh tng S C 50 2C 51 22C 52 25C 55 .
.................................................................................................................................................................
.................................................................................................................................................................
ỏp s: S 35. .....................................................................................................................................
33. Tớnh tng S 4 0C 80 41C 81 42C 82 4 8C 88 .
.................................................................................................................................................................
.................................................................................................................................................................
ỏp s: S 58. .....................................................................................................................................
34. Tỡm n tha món 3n C n0 3n 1C n1 3n 2C n2 3n 3C n3 (1)n C nn 2048.
.................................................................................................................................................................
.................................................................................................................................................................
ỏp s: n 11. ....................................................................................................................................
35. Tỡm n tha món C n1 C n2 C nn 1 C nn 4095.
.................................................................................................................................................................
.................................................................................................................................................................
ỏp s: n 12. ....................................................................................................................................

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Trang - 140 -


ĐỀ CƯƠNG HỌC TẬP LỚP 11

SỞ GD & ĐT TP. HỒ CHÍ MINH
TRUNG TÂM THẾ VINH – 45 LÊ THÚC HOẠCH – Q. TÂN PHÚ

Mơn: Tốn, Năm học: 2019 – 2020

Dạng tồn chẵn hoặc tồn lẻ
2k 1

 Trong biểu thức có C n0  C n2k  ... (tồn chẵn) hoặc C n1  C n

 ... (tồn lẻ) thì đó là

dấu hiệu nhận dạng khai triển hai biểu thức dạng (a  b)n và (a  b)n khi chọn a, b rồi cộng
lại (khi tồn chẵn) hoặc trừ đi (khi tồn lẻ) theo từng vế.
36. Tìm số ngun dương n thỏa mãn: C 20n  C 22n  C 24n  C 26n        C 22nn  512.

 Suy luận: Đây là dạng tồn chẵn, sẽ khai triển 2 nhị thức (a  b)2n và (a  b)2n rồi cộng lại.

 Khơng có số mũ của số nào giảm  Chọn a  1.
 Khơng có số mũ của số nào tăng  Chọn b  1.
Lời giải tham khảo
Xét (1  1)2n 

2n

C

.12n k .1k

 C 20n  C 21n  C 22n  C 23n  C 24n      C 22nn

(1)

Xét (1  1)2n   C 2kn .12n k .(1)k  C 20n  C 21n  C 22n  C 23n  C 24n      C 22nn

(2)

k 0

k
2n

2n

k 0

Lấy (1)  (2)  22n  02n  2.(C 20n  C 22n  C 24n  C 26n      C 22nn )

 22n  2.512  22n  1024  22n  210  2n  10  n  5.
37. Tìm số ngun dương n thỏa mãn: C 21n 1  C 23n 1  C 25n 1      C 22nn11  1024.
.................................................................................................................................................................
.................................................................................................................................................................
.................................................................................................................................................................
.................................................................................................................................................................
.................................................................................................................................................................
.................................................................................................................................................................
Đáp số: n  5. .......................................................................................................................................
2
4
6
8
1006
38. Tìm số ngun dương n thỏa mãn: C 2014
 C 2014
 C 2014
 C 2014
     C 2014
 2503n  1.

.................................................................................................................................................................
.................................................................................................................................................................
.................................................................................................................................................................
.................................................................................................................................................................
.................................................................................................................................................................
.................................................................................................................................................................
.................................................................................................................................................................
Đáp số: n  4. .......................................................................................................................................

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Trang - 141 -


S GD & T TP. H CH MINH
TRUNG TM TH VINH 45 Lấ THC HOCH Q. TN PH

ẹE CệễNG HOẽC TAP LễP 11
Mụn: Toỏn, Nm hc: 2019 2020

39. Chng minh: C 20n C 22n C 22nn C 21n C 23n C 22nn 1 22n 1.
.................................................................................................................................................................
.................................................................................................................................................................
.................................................................................................................................................................
.................................................................................................................................................................
.................................................................................................................................................................
.................................................................................................................................................................
.................................................................................................................................................................
.................................................................................................................................................................
0
2
4
100
40. Tớnh tng: S C 100
C 100
C 100
C 100
.

.................................................................................................................................................................
.................................................................................................................................................................
.................................................................................................................................................................
.................................................................................................................................................................
.................................................................................................................................................................
S: S 299. ...........................................................................................................................................
0
2
4
2000
41. Tớnh tng: S C 2001
32C 2001
34C 2001
32020C 2001
.

.................................................................................................................................................................
.................................................................................................................................................................
.................................................................................................................................................................
.................................................................................................................................................................
S: S 42001 22001. ............................................................................................................................
42.



Tỡm s t nhiờn n tha món: C 20n 32C 22n 34C 24n 32n C 22nn 215 (216 1).

.................................................................................................................................................................
.................................................................................................................................................................
.................................................................................................................................................................
.................................................................................................................................................................
.................................................................................................................................................................
.................................................................................................................................................................
S: n 8. .............................................................................................................................................

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Trang - 142 -


ẹE CệễNG HOẽC TAP LễP 11

S GD & T TP. H CH MINH
TRUNG TM TH VINH 45 Lấ THC HOCH Q. TN PH

Mụn: Toỏn, Nm hc: 2019 2020

Nhúm bi toỏn tớnh tng hoc chng minh da vo tớnh cht hoc bin i (nõng cao)
43.



Tớnh tng: S

1
1
1
1




2!.2012! 4!.2010!
2012!.2! 2014 !

Suy lun: Da vo cụng thc t hp C nk

n!
, cú: k (n k ) n nờn s phõn tớch
k !.(n k )!

1
1
2
v gi cho ta nhõn thờm hai v cho 2014! s a c v C 2014
.

2!.2012! 2!.(2014 2)!
Li gii tham kho
Ta cú: S

1
1
1
1



2!.2012! 4!.2010!
2012!.2! 2014 !

2014!.S

2014 !
2014 !
2014!
2014!



2!.2012! 4!.2010!
2012!.2! 2014!

2014!.S

2014!
2014!
2014!
2014!



2!.(2014 2)! 4!.(2014 4)!
2012!.(2014 2)! 2014!.(2014 2014)!

2
4
6
2012
2014
2014 !.S C 2014
C 2014
C 2014
C 2014
C 2014
.

Xột ...........................................................................................................................................................
.................................................................................................................................................................
.................................................................................................................................................................
.................................................................................................................................................................

22013 1
ỏp s: S
. ..........................................................................................................................
2014 !
44.



Tớnh tng: S

1
1
1
1




2019! 3!.2017 !
2017 !.3! 2020!

.................................................................................................................................................................
.................................................................................................................................................................
.................................................................................................................................................................
.................................................................................................................................................................
.................................................................................................................................................................
.................................................................................................................................................................
.................................................................................................................................................................
.................................................................................................................................................................
.................................................................................................................................................................
ỏp s: S

22019 1
. ..........................................................................................................................
2020!

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Trang - 143 -


ẹE CệễNG HOẽC TAP LễP 11

S GD & T TP. H CH MINH
TRUNG TM TH VINH 45 Lấ THC HOCH Q. TN PH

45.



Tớnh tng: S

Ta cú: S



0
C 2013

1
2013



1



1

1
C 2013

2





1
C 2013

2

2
C 2013

3

2013!



2
C 2013

3





2013
C 2013

2014

2013

2013
C 2013

2014

2013


k 0

1



k
C 2013

1k

2013

1
k
C 2013
k 0 1 k



2014.2013!

1 k k !.(2013 k )! 2014 (1 k ).k !.(2013 k )!
k 0



0
C 2013

Mụn: Toỏn, Nm hc: 2019 2020

k 0

2013

1
2014!
2014 (k 1)!.[2014 (k 1)]!
k 0

2013

1

2014 C
k 0

k 1
2014

1
1
2
2014
C 2014
C 2014
C 2014
.
2014





.................................................................................................................................................................
.................................................................................................................................................................
.................................................................................................................................................................

22014 1
ỏp s: S
. ..........................................................................................................................
2014
46.



Chng minh: k 2C nk n(n 1)C nk22 nC nk11, vi k, n l s nguyờn tha 2 k n.

1
2
3
2013
Tớnh tng: S 12.C 2013
22.C 2013
32.C 2013
20132.C 2013
.

Ta cú: k 2C nk k .k .C nk k .[(k 1) 1].C nk k (k 1).C nk k .C nk

.............................................................................................................................................................
.................................................................................................................................................................
.................................................................................................................................................................
.................................................................................................................................................................
.................................................................................................................................................................
.................................................................................................................................................................
.................................................................................................................................................................
.................................................................................................................................................................
.................................................................................................................................................................
. ................................................................................................................................................................
.................................................................................................................................................................
.................................................................................................................................................................
.................................................................................................................................................................
.................................................................................................................................................................
ỏp s: S 2013.2014.22011. ...............................................................................................................

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Trang - 144 -


ẹE CệễNG HOẽC TAP LễP 11

S GD & T TP. H CH MINH
TRUNG TM TH VINH 45 Lấ THC HOCH Q. TN PH

47.



Mụn: Toỏn, Nm hc: 2019 2020

Chng minh: (C n0 )2 (C n1 )2 (C nn )2 C 2nn vi n 2, n .

Suy lun: Ta cú (1 x )2n (1 x )n (x 1)n nờn suy ngh n vic khai trin (1 x )2n v khai
trin tớch (1 x )n .(x 1)n , sau ú so sỏnh h s x n vi nhau s a n pcm.
Li gii tham kho
2n

Xột khai trin: P (x ) (1 x )2n C 2kn x k cú h s ca x n l C 2nn .
k 0

Xột khai trin P (x ) (1 x )n (x 1)n

x n l

n

(Cnk )2 .x n . Suy ra:
k 0

48.



n

(C
k 0

n
n n

k
k
k n k
C
.
x
.
C
x

n n
(C nk )2 .x n cú h s ca
k 0
k 0
k 0 k 0

n

k 2
n

) .x n C 2nn (C n0 )2 (C n1 )2 (C nn )2 C 2nn (pcm).

0
1
2
2020 2
)2 (C 2020
)2 (C 2020
)2 (C 2020
).
Tớnh tng: S (C 2020

.................................................................................................................................................................
.................................................................................................................................................................
.................................................................................................................................................................
.................................................................................................................................................................
2020
S: S C 4040 . .......................................................................................................................................

49.



C 0 2 C 1 2
C n 2 C n 1 1


Cho s t nhiờn n 2, chng minh: n n n 2n 2 2
1
(n 1)
2
n 1

2
2
2
2
k
n
n
C 0 C 1
C n
2
C
1
n!






n
n
n
n



Ta cú:



1 2
k 1
k 0
k 0 k 1 k !.(n k )!
n 1


2

1

n !.(n 1)





n

1
k
!.(
k

1).(
n

k
)!
k 0

n

2

1


C nk 11


k 0 n 1

n

Suy ra VT

n

1

(n 1)
k 0

2

1

(n 1)!
n 1 (k 1)!.[(n 1) (k 1)!
k 0

n

2

(C nk 11 )2 .

1
(C 1 )2 (C 1 )2 (C n 1 )2
n 1
n 1
n 1
2

(n 1)

.................................................................................................................................................................
.................................................................................................................................................................
.................................................................................................................................................................
.................................................................................................................................................................
.................................................................................................................................................................

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Trang - 145 -


ẹE CệễNG HOẽC TAP LễP 11

S GD & T TP. H CH MINH
TRUNG TM TH VINH 45 Lấ THC HOCH Q. TN PH

50.



Mụn: Toỏn, Nm hc: 2019 2020

Tỡm s nguyờn dng n tha món: 2C n0 5C n1 8C n2 (3n 2)C nn 1600.
n

Hng dn: 2C n0 5C n1 8C n2 (3n 2)C nn (3k 2)C nk

S: n 7.

k 0

51.



Tớnh tng: S

Hng dn: S

C 1212
11.12
2014



C 1312
11.12



C 1412
11.12

C k12

(k 1).k

k 12

52.





C n102
132

12
C 2013

2012.2013



12
C 2014

2013.2014



S: S

n

n
1
1
C nk
C nk 11.
k 0 k 1
k 0 n 1



11
C 2013

132



1
1
1
1
1023
Tỡm s nguyờn dng n tha món: C n0 C n1 C n2 C n3
C nn

2
3
4
n 1
n 1

Hng dn: VT
53.



S: n 9.

2
3
2013
2.3.C 2013
2012.2013.C 2013
.
Tớnh tng: S 1.2.C 2013
2013

k
k 2
Hng dn: S (k 1).k .C 2013
2012.2013.C 2011
.

S: 2012.2013.22011.

k 2

54.



0
1
2
3
4
2012
2C 2012
3C 2012
4C 2012
5C 2012
2013C 2012
.
Tớnh tng: S C 2012

Hng dn: S

2012

k

(k 1)C 2012
k 0

55.



2010

C
k 0



n

k 0

2010k
k
C 2011
2011C 2010
.
k

Tớnh tng: S

0
A2013

0!

3
k





1
A2013

1!



n



k 3



1
1
1
1
89
3 3 3

3
30
C3 C4 C5
Cn
1

1



3 (k 1)(k 2) k(k 1).

2
A2013

2!





2013
A2013

2013!

Ank
n!
Hng dn: C

v 0! 1.
k !(n k )!
k!


Tỡm s nguyờn dng n 2 tha món:

Hng dn: C nk
59.



S: n 10.



k
n

58.

S: 1007.22012.

S: n 2010.

k
2011

1

C
k 3



k 1

Tỡm s nguyờn dng n 3 tha món:

Hng dn: VT

57.

2012

k 1
k
C 2012
.
2012C 2011

0
2010
1
2009
k
2010k
2010 0
C 2011
C 2011
C 2010
C 2011
C 2011
C 2011
C 1 2011.2n.
Tỡm n tha: C 2011
k

Hng dn: VT
56.

2011

S: S 22013.

1
1
1
1
2013
2 2 2

2
2014
A2
A3
A4
An

Ak
n!
1
1
1 1
1
1
1

n k

VT








k !(n k )!
k!
2 C 22 C 32 C 42
An
k !C nk
C n2

11
1
10 1
11 0
C 20
C1210 C 202 C129 C 20
C 12 C 20
C12 .
Tớnh tng: S C 200 C12

Hng dn: So sỏnh h s x 11 trong (1 x )32 v (1 x )20 (1 x )12 .

Biên soạn và giảng dạy: Ths. Lê Văn Đoàn 0933.755.607 0929.031.789

11
.
S: S C 32

Trang - 146 -


S GD & T TP. H CH MINH
TRUNG TM TH VINH 45 Lấ THC HOCH Q. TN PH

ẹE CệễNG HOẽC TAP LễP 11
Mụn: Toỏn, Nm hc: 2019 2020

BI TP V NH 3
Cõu 1. Xột (1 2x )20 a 0 a1x a 2x 2 a 20x 20 . Giỏ tr a 0 a1 a 2 a 20 bng
A. 1.

B. 320.

..............................................................................................

C. 0.

D. 1.

..............................................................................................

Cõu 2. Tớnh tng cỏc h s trong khai trin (1 2x )2018 .
A. 1.

B. 1.

..............................................................................................

C. 2018.

D. 2018.

..............................................................................................

Cõu 3. Xột khai trin a thc (1 2x 3x 2 )10 a 0 a1x a2x 2 a20x 20 . Giỏ tr ca
tng S a 0 2a1 4a 2 220 a 20 bng
A. 1510.

B. 1710.

..............................................................................................

C. 710.

D. 1720.

..............................................................................................

Cõu 4. Cho a thc P (x ) (x 2)2017 (3 2x )2018 a2018x 2018 a 2017 x 2017 a1x a 0 .
Khi ú S a 2018 a 2017 a1 a 0 bng
A. 0.

B. 1.

..............................................................................................

C. 2018.

D. 2017.

..............................................................................................

1
2
3
2016
Cõu 5. Tng C 2016
C 2016
C 2016
C 2016
bng

A. 42016.

B. 22016 1.

..............................................................................................

C. 42016 1.

D. 22016 1.

..............................................................................................

1
Cõu 6. Tớnh tng S C 100 2.C 10
22.C 102 210.C 1010 .

A. S 210.

B. S 410.

..............................................................................................

C. S 310.

D. S 311.

..............................................................................................

Cõu 7. Tỡm s nguyờn dng n tha món C n0 2C n1 22C n2 2n C nn 14348907.
A. n 15.

B. n 14.

..............................................................................................

C. n 10.

D. n 11.

..............................................................................................

Cõu 8. Tỡm s nguyờn dng n tha: 3n C n0 3n 1C n1 3n 2C n2 (1)n C nn 2048.
A. n 8.

B. n 9.

..............................................................................................

C. n 10.

D. n 11.

..............................................................................................

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Trang - 147 -


S GD & T TP. H CH MINH
TRUNG TM TH VINH 45 Lấ THC HOCH Q. TN PH

ẹE CệễNG HOẽC TAP LễP 11
Mụn: Toỏn, Nm hc: 2019 2020

Cõu 9. Tớnh tng S 5n C n0 5n 1.3.C nn 1 32.5n 2C nn 2 3n C n0 .
A. 28n.

B. 1 8n.

..............................................................................................

C. 8n1.

D. 8n.

..............................................................................................

Cõu 10. Tng tt c cỏc h s ca khai trin (x y )20 bng bao nhiờu ?
A. 77520.

B. 1860480.

..............................................................................................

C. 1048576.

D. 81920.

..............................................................................................

Cõu 11. Trong khai trin nh thc (3 0, 02)7 , tỡm tng s ba s hng u tiờn ?
A. 2289, 3283.

B. 2291,1012.

..............................................................................................

C. 2275, 93801.

D. 2291,1141.

..............................................................................................

Cõu 12. Tng C 20n C 22n C 24n C 22nn bng
..............................................................................................
n2

A. 2

n1

.

B. 2

C. 22n2.

.

D. 22n1.

..............................................................................................
..............................................................................................
..............................................................................................

Cõu 13. Tỡm s nguyờn dng n tha món C 21n 1 C 23n 1 C 22nn11 1024.
..............................................................................................
A. n 10.

B. n 5.

C. n 9.

D. n 11.

..............................................................................................
..............................................................................................
..............................................................................................

Cõu 14. Tỡm s nguyờn dng n tha món C 20n 1 C 22n 1 C 24n 1 C 22nn1 1024.
..............................................................................................
A. n 6.

B. n 10.

C. n 5.

D. n 9.

..............................................................................................
..............................................................................................
..............................................................................................

1
3
5
2017
Cõu 15. Tng T C 2017
C 2017
C 2017
C 2017
bng

..............................................................................................
2017

A. 2

C. 22017.

1.

2016

B. 2

.

D. 22016 1.

..............................................................................................
..............................................................................................
..............................................................................................

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Trang - 148 -


ẹE CệễNG HOẽC TAP LễP 11

S GD & T TP. H CH MINH
TRUNG TM TH VINH 45 Lấ THC HOCH Q. TN PH

Cõu 16.



Tng S C n0 .C 21n C n1 .C 22n C nn .C 2nn1 bng

2n
3n

B. C .

C. C 3nn1 .

D. C 32nn 1 .


..............................................................................................
..............................................................................................
..............................................................................................

Tớnh tng P (C n0 )2 (C n1 )2 (C nn )2 theo n.

n
n

..............................................................................................

2
n

A. C .

B. C .

C. C 2nn .

D. C 22nn .

Cõu 18.

..............................................................................................

n
3n

A. C .

Cõu 17.



Mụn: Toỏn, Nm hc: 2019 2020

..............................................................................................
..............................................................................................
..............................................................................................

Tớnh tng S C n1 2C n2 3C n3 nC nn .

A. 4n.2n 1.

B. n.2n 1.

C. 3n.2n 1.

D. 2n.2n 1.

..............................................................................................
..............................................................................................
..............................................................................................
..............................................................................................

Cõu 19.



Tỡm s nguyờn dng n tha món 2C n0 5C n1 8C n2 (3n 2)C nn 1600.

A. n 5.

B. n 7.

C. n 10.

D. n 8.

..............................................................................................
..............................................................................................
..............................................................................................
..............................................................................................

Cõu 20.



1
1
1
Tớnh cỏc tng sau: S1 C n0 C n1 C n2
C nn .
2
3
n 1

2n 1 1
A.

n 1

..............................................................................................

2n 1 1
B.

n 1

..............................................................................................

2n 1 1
2n 1 1
C.
1. D.
1.
n 1
n 1

..............................................................................................
..............................................................................................

P N V NH S 03
1.A

2.B

3.B

4.A

5.D

6.C

7.A

8.D

9.D

10.C

11.B

12.D

13.B

14.C

15.B

16.C

17.C

18.B

19.B

20.B

Biên soạn và giảng dạy: Ths. Lê Văn Đoàn 0933.755.607 0929.031.789

Trang - 149 -


ẹE CệễNG HOẽC TAP LễP 11

S GD & T TP. H CH MINH
TRUNG TM TH VINH 45 Lấ THC HOCH Q. TN PH

Mụn: Toỏn, Nm hc: 2019 2020

Dng toỏn 3. Tỡm h s hoc s hng dng cú iu kin (kt hp gia dng 1 & 2)
n


2
60. Tỡm s hng khụng cha x trong khai trin x 3 2 , bit C n1 C n2 55.

x
iu kin: n 2 v n .
Ta cú: C n1 C n2 55

n!
n!

55 n 2 n 110 0
(n 1)! (n 2)!2!

n 10

n 11 (L).


k

10



2
k 3(10k ) 2
k k 305k
Vi n 10 ta cú x 3 2 vi s hng tng quỏt: Tk 1 C 10
x
.
.
2 C 10 2 x

x
x
S hng khụng cha x ng vi k tha 30 5k 0 k 6.
Vy s hng khụng cha x l C 106 2 6 13440.
n

61. Tỡm s hng cha x

10


1
trong khai trin x 3 2 , bit C n4 13C n2 .

x

.................................................................................................................................................................
.................................................................................................................................................................
.................................................................................................................................................................
.................................................................................................................................................................
S: C104 (1)4 .x 10 210x 10 . ..................................................................................................................
n

62. Tỡm h s ca x

20


1
trong khai trin nh thc Newton x 4 , bit An2 3n 440.

2

.................................................................................................................................................................
.................................................................................................................................................................
.................................................................................................................................................................
.................................................................................................................................................................
15 15
2 . .................................................................................................................................
S: (1)15C 20
3
2
1
63. Tỡm s hng cha x 8 trong khai trin (x 2 2)n , bit An 8Cn C n 49.

.................................................................................................................................................................
.................................................................................................................................................................
.................................................................................................................................................................
.................................................................................................................................................................
S: 280x 8 . .............................................................................................................................................

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Trang - 150 -


ẹE CệễNG HOẽC TAP LễP 11

S GD & T TP. H CH MINH
TRUNG TM TH VINH 45 Lấ THC HOCH Q. TN PH

Mụn: Toỏn, Nm hc: 2019 2020

n


1
64. Tỡm s hng cha x trong khai trin x 3 2 , x 0, bit Cn0 C n1 C n2 11.

x
2

.................................................................................................................................................................
.................................................................................................................................................................
.................................................................................................................................................................
.................................................................................................................................................................
S: 6x 2 . ..................................................................................................................................................
n

2

65. Tỡm h s ca x trong khai trin x 3 , x 0, bit Cnn46 n.An2 454.
x

4

.................................................................................................................................................................
.................................................................................................................................................................
.................................................................................................................................................................
.................................................................................................................................................................
S: 1792. ............................................................................................................................................
n


3
66. Tỡm s hng khụng cha x trong khai trin x 3 , bit An21 Cn21 18P3 .

x
.................................................................................................................................................................
.................................................................................................................................................................
.................................................................................................................................................................
.................................................................................................................................................................
S: 252. .................................................................................................................................................
n

67. Tỡm h s ca x

11


n
trong khai trin x x n 8 , bit Cnn3 C n21 Cn11C nn32 .

3x
3

.................................................................................................................................................................
.................................................................................................................................................................
.................................................................................................................................................................
.................................................................................................................................................................
.................................................................................................................................................................
.................................................................................................................................................................
.................................................................................................................................................................
S: 32440320. .......................................................................................................................................

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Trang - 151 -


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