12x1 t08 05 binomial coefficients (2010)
TRANSCRIPT
![Page 1: 12X1 T08 05 binomial coefficients (2010)](https://reader034.vdocuments.us/reader034/viewer/2022052412/5591a2211a28ab9a268b476b/html5/thumbnails/1.jpg)
Relationships Between Binomial Coefficients
![Page 2: 12X1 T08 05 binomial coefficients (2010)](https://reader034.vdocuments.us/reader034/viewer/2022052412/5591a2211a28ab9a268b476b/html5/thumbnails/2.jpg)
Relationships Between Binomial Coefficients
Binomial Theorem
![Page 3: 12X1 T08 05 binomial coefficients (2010)](https://reader034.vdocuments.us/reader034/viewer/2022052412/5591a2211a28ab9a268b476b/html5/thumbnails/3.jpg)
Relationships Between Binomial Coefficients
Binomial Theorem
n
k
kk
nn xCx0
1
![Page 4: 12X1 T08 05 binomial coefficients (2010)](https://reader034.vdocuments.us/reader034/viewer/2022052412/5591a2211a28ab9a268b476b/html5/thumbnails/4.jpg)
Relationships Between Binomial Coefficients
Binomial Theorem
n
k
kk
nn xCx0
1
nn
nkk
nnnn xCxCxCxCC 2210
![Page 5: 12X1 T08 05 binomial coefficients (2010)](https://reader034.vdocuments.us/reader034/viewer/2022052412/5591a2211a28ab9a268b476b/html5/thumbnails/5.jpg)
Relationships Between Binomial Coefficients
Binomial Theorem
n
k
kk
nn xCx0
1
nn
nkk
nnnn xCxCxCxCC 2210
e.g. (i) Find the values of;
n
kk
nC1
a)
![Page 6: 12X1 T08 05 binomial coefficients (2010)](https://reader034.vdocuments.us/reader034/viewer/2022052412/5591a2211a28ab9a268b476b/html5/thumbnails/6.jpg)
Relationships Between Binomial Coefficients
Binomial Theorem
n
k
kk
nn xCx0
1
nn
nkk
nnnn xCxCxCxCC 2210
e.g. (i) Find the values of;
n
kk
nC1
a)
n
k
kk
nn xCx0
1
![Page 7: 12X1 T08 05 binomial coefficients (2010)](https://reader034.vdocuments.us/reader034/viewer/2022052412/5591a2211a28ab9a268b476b/html5/thumbnails/7.jpg)
Relationships Between Binomial Coefficients
Binomial Theorem
n
k
kk
nn xCx0
1
nn
nkk
nnnn xCxCxCxCC 2210
e.g. (i) Find the values of;
n
kk
nC1
a)
let x = 1;
n
k
kk
nn xCx0
1
n
k
kk
nn C0
111
![Page 8: 12X1 T08 05 binomial coefficients (2010)](https://reader034.vdocuments.us/reader034/viewer/2022052412/5591a2211a28ab9a268b476b/html5/thumbnails/8.jpg)
Relationships Between Binomial Coefficients
Binomial Theorem
n
k
kk
nn xCx0
1
nn
nkk
nnnn xCxCxCxCC 2210
e.g. (i) Find the values of;
n
kk
nC1
a)
let x = 1;
n
k
kk
nn xCx0
1
n
k
kk
nn C0
111
n
kk
nn C0
2
![Page 9: 12X1 T08 05 binomial coefficients (2010)](https://reader034.vdocuments.us/reader034/viewer/2022052412/5591a2211a28ab9a268b476b/html5/thumbnails/9.jpg)
Relationships Between Binomial Coefficients
Binomial Theorem
n
k
kk
nn xCx0
1
nn
nkk
nnnn xCxCxCxCC 2210
e.g. (i) Find the values of;
n
kk
nC1
a)
let x = 1;
n
k
kk
nn xCx0
1
n
k
kk
nn C0
111
n
kk
nn C0
2
n
kk
nnn CC1
02
![Page 10: 12X1 T08 05 binomial coefficients (2010)](https://reader034.vdocuments.us/reader034/viewer/2022052412/5591a2211a28ab9a268b476b/html5/thumbnails/10.jpg)
Relationships Between Binomial Coefficients
Binomial Theorem
n
k
kk
nn xCx0
1
nn
nkk
nnnn xCxCxCxCC 2210
e.g. (i) Find the values of;
n
kk
nC1
a)
let x = 1;
n
k
kk
nn xCx0
1
n
k
kk
nn C0
111
n
kk
nn C0
2
n
kk
nnn CC1
02
01
2 CC nnn
kk
n
![Page 11: 12X1 T08 05 binomial coefficients (2010)](https://reader034.vdocuments.us/reader034/viewer/2022052412/5591a2211a28ab9a268b476b/html5/thumbnails/11.jpg)
Relationships Between Binomial Coefficients
Binomial Theorem
n
k
kk
nn xCx0
1
nn
nkk
nnnn xCxCxCxCC 2210
e.g. (i) Find the values of;
n
kk
nC1
a)
let x = 1;
n
k
kk
nn xCx0
1
n
k
kk
nn C0
111
n
kk
nn C0
2
n
kk
nnn CC1
02
01
2 CC nnn
kk
n
121
nn
kk
nC
![Page 12: 12X1 T08 05 binomial coefficients (2010)](https://reader034.vdocuments.us/reader034/viewer/2022052412/5591a2211a28ab9a268b476b/html5/thumbnails/12.jpg)
7531b) CCCC nnnn
![Page 13: 12X1 T08 05 binomial coefficients (2010)](https://reader034.vdocuments.us/reader034/viewer/2022052412/5591a2211a28ab9a268b476b/html5/thumbnails/13.jpg)
7531b) CCCC nnnn
n
k
kk
nn xCx0
1
![Page 14: 12X1 T08 05 binomial coefficients (2010)](https://reader034.vdocuments.us/reader034/viewer/2022052412/5591a2211a28ab9a268b476b/html5/thumbnails/14.jpg)
7531b) CCCC nnnn
n
k
kk
nn xCx0
1
55
44
33
2210 xCxCxCxCxCC nnnnnn
![Page 15: 12X1 T08 05 binomial coefficients (2010)](https://reader034.vdocuments.us/reader034/viewer/2022052412/5591a2211a28ab9a268b476b/html5/thumbnails/15.jpg)
7531b) CCCC nnnn
let x = 1;
n
k
kk
nn xCx0
1
55
44
33
2210 xCxCxCxCxCC nnnnnn
54321011 CCCCCC nnnnnnn
![Page 16: 12X1 T08 05 binomial coefficients (2010)](https://reader034.vdocuments.us/reader034/viewer/2022052412/5591a2211a28ab9a268b476b/html5/thumbnails/16.jpg)
7531b) CCCC nnnn
let x = 1;
n
k
kk
nn xCx0
1
55
44
33
2210 xCxCxCxCxCC nnnnnn
54321011 CCCCCC nnnnnnn
1 2 543210 CCCCCC nnnnnnn
![Page 17: 12X1 T08 05 binomial coefficients (2010)](https://reader034.vdocuments.us/reader034/viewer/2022052412/5591a2211a28ab9a268b476b/html5/thumbnails/17.jpg)
7531b) CCCC nnnn
let x = 1;
n
k
kk
nn xCx0
1
55
44
33
2210 xCxCxCxCxCC nnnnnn
54321011 CCCCCC nnnnnnn
1 2 543210 CCCCCC nnnnnnn
let x = -1; 54321011 CCCCCC nnnnnnn
![Page 18: 12X1 T08 05 binomial coefficients (2010)](https://reader034.vdocuments.us/reader034/viewer/2022052412/5591a2211a28ab9a268b476b/html5/thumbnails/18.jpg)
7531b) CCCC nnnn
let x = 1;
n
k
kk
nn xCx0
1
55
44
33
2210 xCxCxCxCxCC nnnnnn
54321011 CCCCCC nnnnnnn
1 2 543210 CCCCCC nnnnnnn
let x = -1; 54321011 CCCCCC nnnnnnn
2 0 543210 CCCCCC nnnnnn
![Page 19: 12X1 T08 05 binomial coefficients (2010)](https://reader034.vdocuments.us/reader034/viewer/2022052412/5591a2211a28ab9a268b476b/html5/thumbnails/19.jpg)
7531b) CCCC nnnn
let x = 1;
n
k
kk
nn xCx0
1
55
44
33
2210 xCxCxCxCxCC nnnnnn
54321011 CCCCCC nnnnnnn
1 2 543210 CCCCCC nnnnnnn
let x = -1; 54321011 CCCCCC nnnnnnn
2 0 543210 CCCCCC nnnnnn
subtract (2) from (1)
![Page 20: 12X1 T08 05 binomial coefficients (2010)](https://reader034.vdocuments.us/reader034/viewer/2022052412/5591a2211a28ab9a268b476b/html5/thumbnails/20.jpg)
7531b) CCCC nnnn
let x = 1;
n
k
kk
nn xCx0
1
55
44
33
2210 xCxCxCxCxCC nnnnnn
54321011 CCCCCC nnnnnnn
1 2 543210 CCCCCC nnnnnnn
let x = -1; 54321011 CCCCCC nnnnnnn
2 0 543210 CCCCCC nnnnnn
subtract (2) from (1)
531 2222 CCC nnnn
![Page 21: 12X1 T08 05 binomial coefficients (2010)](https://reader034.vdocuments.us/reader034/viewer/2022052412/5591a2211a28ab9a268b476b/html5/thumbnails/21.jpg)
7531b) CCCC nnnn
let x = 1;
n
k
kk
nn xCx0
1
55
44
33
2210 xCxCxCxCxCC nnnnnn
54321011 CCCCCC nnnnnnn
1 2 543210 CCCCCC nnnnnnn
let x = -1; 54321011 CCCCCC nnnnnnn
2 0 543210 CCCCCC nnnnnn
subtract (2) from (1)
531 2222 CCC nnnn
531
12 CCC nnnn
![Page 22: 12X1 T08 05 binomial coefficients (2010)](https://reader034.vdocuments.us/reader034/viewer/2022052412/5591a2211a28ab9a268b476b/html5/thumbnails/22.jpg)
n
kk
nCk1
c)
![Page 23: 12X1 T08 05 binomial coefficients (2010)](https://reader034.vdocuments.us/reader034/viewer/2022052412/5591a2211a28ab9a268b476b/html5/thumbnails/23.jpg)
n
kk
nCk1
c)
n
k
kk
nn xCx0
1
![Page 24: 12X1 T08 05 binomial coefficients (2010)](https://reader034.vdocuments.us/reader034/viewer/2022052412/5591a2211a28ab9a268b476b/html5/thumbnails/24.jpg)
n
kk
nCk1
c)
Differentiate both sides
n
k
kk
nn xCx0
1
![Page 25: 12X1 T08 05 binomial coefficients (2010)](https://reader034.vdocuments.us/reader034/viewer/2022052412/5591a2211a28ab9a268b476b/html5/thumbnails/25.jpg)
n
kk
nCk1
c)
Differentiate both sides
n
k
kk
nn xCx0
1
1
0
11
kn
kk
nn xCkxn
![Page 26: 12X1 T08 05 binomial coefficients (2010)](https://reader034.vdocuments.us/reader034/viewer/2022052412/5591a2211a28ab9a268b476b/html5/thumbnails/26.jpg)
n
kk
nCk1
c)
Differentiate both sides
n
k
kk
nn xCx0
1
1
0
11
kn
kk
nn xCkxn
let x = 1;
n
kk
nn Ckn0
111
![Page 27: 12X1 T08 05 binomial coefficients (2010)](https://reader034.vdocuments.us/reader034/viewer/2022052412/5591a2211a28ab9a268b476b/html5/thumbnails/27.jpg)
n
kk
nCk1
c)
Differentiate both sides
n
k
kk
nn xCx0
1
1
0
11
kn
kk
nn xCkxn
let x = 1;
n
kk
nn Ckn0
111
n
kk
nnn CkCn1
01 02
![Page 28: 12X1 T08 05 binomial coefficients (2010)](https://reader034.vdocuments.us/reader034/viewer/2022052412/5591a2211a28ab9a268b476b/html5/thumbnails/28.jpg)
n
kk
nCk1
c)
Differentiate both sides
n
k
kk
nn xCx0
1
1
0
11
kn
kk
nn xCkxn
let x = 1;
n
kk
nn Ckn0
111
n
kk
nnn CkCn1
01 02
1
12
nn
kk
n nCk
![Page 29: 12X1 T08 05 binomial coefficients (2010)](https://reader034.vdocuments.us/reader034/viewer/2022052412/5591a2211a28ab9a268b476b/html5/thumbnails/29.jpg)
n
k
knk
kC
0 11 d)
![Page 30: 12X1 T08 05 binomial coefficients (2010)](https://reader034.vdocuments.us/reader034/viewer/2022052412/5591a2211a28ab9a268b476b/html5/thumbnails/30.jpg)
n
k
knk
kC
0 11 d)
n
k
kk
nn xCx0
1
![Page 31: 12X1 T08 05 binomial coefficients (2010)](https://reader034.vdocuments.us/reader034/viewer/2022052412/5591a2211a28ab9a268b476b/html5/thumbnails/31.jpg)
n
k
knk
kC
0 11 d)
Integrate both sides
n
k
kk
nn xCx0
1
![Page 32: 12X1 T08 05 binomial coefficients (2010)](https://reader034.vdocuments.us/reader034/viewer/2022052412/5591a2211a28ab9a268b476b/html5/thumbnails/32.jpg)
n
k
knk
kC
0 11 d)
Integrate both sides
n
k
kk
nn xCx0
1
11
1 1
0
1
kxCK
nx kn
kk
nn
![Page 33: 12X1 T08 05 binomial coefficients (2010)](https://reader034.vdocuments.us/reader034/viewer/2022052412/5591a2211a28ab9a268b476b/html5/thumbnails/33.jpg)
n
k
knk
kC
0 11 d)
Integrate both sides
n
k
kk
nn xCx0
1
11
1 1
0
1
kxCK
nx kn
kk
nn
let x = 0; 1
01
01 1
0
1
kCK
n
kn
kk
nn
![Page 34: 12X1 T08 05 binomial coefficients (2010)](https://reader034.vdocuments.us/reader034/viewer/2022052412/5591a2211a28ab9a268b476b/html5/thumbnails/34.jpg)
n
k
knk
kC
0 11 d)
Integrate both sides
n
k
kk
nn xCx0
1
11
1 1
0
1
kxCK
nx kn
kk
nn
let x = 0; 1
01
01 1
0
1
kCK
n
kn
kk
nn
11
n
K
![Page 35: 12X1 T08 05 binomial coefficients (2010)](https://reader034.vdocuments.us/reader034/viewer/2022052412/5591a2211a28ab9a268b476b/html5/thumbnails/35.jpg)
n
k
knk
kC
0 11 d)
Integrate both sides
n
k
kk
nn xCx0
1
11
1 1
0
1
kxCK
nx kn
kk
nn
let x = 0; 1
01
01 1
0
1
kCK
n
kn
kk
nn
11
n
K
let x = -1; 1
11
111 1
0
1
kC
n
kn
kk
nn
![Page 36: 12X1 T08 05 binomial coefficients (2010)](https://reader034.vdocuments.us/reader034/viewer/2022052412/5591a2211a28ab9a268b476b/html5/thumbnails/36.jpg)
n
k
knk
kC
0 11 d)
Integrate both sides
n
k
kk
nn xCx0
1
11
1 1
0
1
kxCK
nx kn
kk
nn
let x = 0; 1
01
01 1
0
1
kCK
n
kn
kk
nn
11
n
K
let x = -1; 1
11
111 1
0
1
kC
n
kn
kk
nn
1
11
1 1
0
nk
Ckn
kk
n
![Page 37: 12X1 T08 05 binomial coefficients (2010)](https://reader034.vdocuments.us/reader034/viewer/2022052412/5591a2211a28ab9a268b476b/html5/thumbnails/37.jpg)
n
k
knk
kC
0 11 d)
Integrate both sides
n
k
kk
nn xCx0
1
11
1 1
0
1
kxCK
nx kn
kk
nn
let x = 0; 1
01
01 1
0
1
kCK
n
kn
kk
nn
11
n
K
let x = -1; 1
11
111 1
0
1
kC
n
kn
kk
nn
1
11
1 1
0
nk
Ckn
kk
n
1
11
10
nk
Ckn
kk
n
![Page 38: 12X1 T08 05 binomial coefficients (2010)](https://reader034.vdocuments.us/reader034/viewer/2022052412/5591a2211a28ab9a268b476b/html5/thumbnails/38.jpg)
nnn
n
xxx
xii2111
identity; theof sidesboth on of tscoefficien theequatingBy
2
2
0 !!2
that;show
nn
kn
n
k
![Page 39: 12X1 T08 05 binomial coefficients (2010)](https://reader034.vdocuments.us/reader034/viewer/2022052412/5591a2211a28ab9a268b476b/html5/thumbnails/39.jpg)
nnn
n
xxx
xii2111
identity; theof sidesboth on of tscoefficien theequatingBy
2
2
0 !!2
that;show
nn
kn
n
k
n
k
kk
nn xCx0
1
![Page 40: 12X1 T08 05 binomial coefficients (2010)](https://reader034.vdocuments.us/reader034/viewer/2022052412/5591a2211a28ab9a268b476b/html5/thumbnails/40.jpg)
nnn
n
xxx
xii2111
identity; theof sidesboth on of tscoefficien theequatingBy
2
2
0 !!2
that;show
nn
kn
n
k
n
k
kk
nn xCx0
1
nn
nnn
nnn
nnnn xCxCxCxCxCC
11
22
2210
![Page 41: 12X1 T08 05 binomial coefficients (2010)](https://reader034.vdocuments.us/reader034/viewer/2022052412/5591a2211a28ab9a268b476b/html5/thumbnails/41.jpg)
nnn
n
xxx
xii2111
identity; theof sidesboth on of tscoefficien theequatingBy
2
2
0 !!2
that;show
nn
kn
n
k
n
k
kk
nn xCx0
1
nn
nnn
nnn
nnnn xCxCxCxCxCC
11
22
2210
nnn xxx 11in oft coefficien
![Page 42: 12X1 T08 05 binomial coefficients (2010)](https://reader034.vdocuments.us/reader034/viewer/2022052412/5591a2211a28ab9a268b476b/html5/thumbnails/42.jpg)
nnn
n
xxx
xii2111
identity; theof sidesboth on of tscoefficien theequatingBy
2
2
0 !!2
that;show
nn
kn
n
k
n
k
kk
nn xCx0
1
nn
nnn
nnn
nnnn xCxCxCxCxCC
11
22
2210
nnn xxx 11in oft coefficien
nn
nnn
nnn
nnnn
nn
nnn
nnn
nnnn
xCxCxCxCxCC
xCxCxCxCxCC
11
22
2210
11
22
2210
![Page 43: 12X1 T08 05 binomial coefficients (2010)](https://reader034.vdocuments.us/reader034/viewer/2022052412/5591a2211a28ab9a268b476b/html5/thumbnails/43.jpg)
nnn
n
xxx
xii2111
identity; theof sidesboth on of tscoefficien theequatingBy
2
2
0 !!2
that;show
nn
kn
n
k
n
k
kk
nn xCx0
1
nn
nnn
nnn
nnnn xCxCxCxCxCC
11
22
2210
nnn xxx 11in oft coefficien
nxnnn
0
01122122 n
xnn
xn
xn
nx
nx
nn nnn
nn
nnn
nnn
nnnn
nn
nnn
nnn
nnnn
xCxCxCxCxCC
xCxCxCxCxCC
11
22
2210
11
22
2210
![Page 44: 12X1 T08 05 binomial coefficients (2010)](https://reader034.vdocuments.us/reader034/viewer/2022052412/5591a2211a28ab9a268b476b/html5/thumbnails/44.jpg)
nnn
n
xxx
xii2111
identity; theof sidesboth on of tscoefficien theequatingBy
2
2
0 !!2
that;show
nn
kn
n
k
n
k
kk
nn xCx0
1
nn
nnn
nnn
nnnn xCxCxCxCxCC
11
22
2210
nnn xxx 11in oft coefficien
nxnnn
01
11
nx
nn
xn
nn
nnn
nnn
nnnn
nn
nnn
nnn
nnnn
xCxCxCxCxCC
xCxCxCxCxCC
11
22
2210
11
22
2210
![Page 45: 12X1 T08 05 binomial coefficients (2010)](https://reader034.vdocuments.us/reader034/viewer/2022052412/5591a2211a28ab9a268b476b/html5/thumbnails/45.jpg)
nnn
n
xxx
xii2111
identity; theof sidesboth on of tscoefficien theequatingBy
2
2
0 !!2
that;show
nn
kn
n
k
n
k
kk
nn xCx0
1
nn
nnn
nnn
nnnn xCxCxCxCxCC
11
22
2210
nnn xxx 11in oft coefficien
nxnnn
01
11
nx
nn
xn 22
22
nx
nn
xn
nn
nnn
nnn
nnnn
nn
nnn
nnn
nnnn
xCxCxCxCxCC
xCxCxCxCxCC
11
22
2210
11
22
2210
![Page 46: 12X1 T08 05 binomial coefficients (2010)](https://reader034.vdocuments.us/reader034/viewer/2022052412/5591a2211a28ab9a268b476b/html5/thumbnails/46.jpg)
nnn
n
xxx
xii2111
identity; theof sidesboth on of tscoefficien theequatingBy
2
2
0 !!2
that;show
nn
kn
n
k
n
k
kk
nn xCx0
1
nn
nnn
nnn
nnnn xCxCxCxCxCC
11
22
2210
nnn xxx 11in oft coefficien
nxnnn
01
11
nx
nn
xn 22
22
nx
nn
xn
01122122 n
xnn
xn
xn
nx
nx
nn nnn
nn
nnn
nnn
nnnn
nn
nnn
nnn
nnnn
xCxCxCxCxCC
xCxCxCxCxCC
11
22
2210
11
22
2210
![Page 47: 12X1 T08 05 binomial coefficients (2010)](https://reader034.vdocuments.us/reader034/viewer/2022052412/5591a2211a28ab9a268b476b/html5/thumbnails/47.jpg)
022110 oft coefficien
nnn
nnn
nnn
nnn
xn
![Page 48: 12X1 T08 05 binomial coefficients (2010)](https://reader034.vdocuments.us/reader034/viewer/2022052412/5591a2211a28ab9a268b476b/html5/thumbnails/48.jpg)
kn
nkn
But
022110 oft coefficien
nnn
nnn
nnn
nnn
xn
![Page 49: 12X1 T08 05 binomial coefficients (2010)](https://reader034.vdocuments.us/reader034/viewer/2022052412/5591a2211a28ab9a268b476b/html5/thumbnails/49.jpg)
kn
nkn
But 2222
210
nnnnn
022110 oft coefficien
nnn
nnn
nnn
nnn
xn
![Page 50: 12X1 T08 05 binomial coefficients (2010)](https://reader034.vdocuments.us/reader034/viewer/2022052412/5591a2211a28ab9a268b476b/html5/thumbnails/50.jpg)
kn
nkn
But 2222
210
nnnnn
n
k kn
0
2
022110 oft coefficien
nnn
nnn
nnn
nnn
xn
![Page 51: 12X1 T08 05 binomial coefficients (2010)](https://reader034.vdocuments.us/reader034/viewer/2022052412/5591a2211a28ab9a268b476b/html5/thumbnails/51.jpg)
kn
nkn
But 2222
210
nnnnn
n
k kn
0
2
nn xx 21in oft coefficien
022110 oft coefficien
nnn
nnn
nnn
nnn
xn
![Page 52: 12X1 T08 05 binomial coefficients (2010)](https://reader034.vdocuments.us/reader034/viewer/2022052412/5591a2211a28ab9a268b476b/html5/thumbnails/52.jpg)
kn
nkn
But 2222
210
nnnnn
n
k kn
0
2
nn xx 21in oft coefficien
nnn xnn
xnn
xn
xnn
x 222
222
22
12
02
1
022110 oft coefficien
nnn
nnn
nnn
nnn
xn
![Page 53: 12X1 T08 05 binomial coefficients (2010)](https://reader034.vdocuments.us/reader034/viewer/2022052412/5591a2211a28ab9a268b476b/html5/thumbnails/53.jpg)
kn
nkn
But 2222
210
nnnnn
n
k kn
0
2
nn xx 21in oft coefficien
nnn xnn
xnn
xn
xnn
x 222
222
22
12
02
1
022110 oft coefficien
nnn
nnn
nnn
nnn
xn
![Page 54: 12X1 T08 05 binomial coefficients (2010)](https://reader034.vdocuments.us/reader034/viewer/2022052412/5591a2211a28ab9a268b476b/html5/thumbnails/54.jpg)
kn
nkn
But 2222
210
nnnnn
n
k kn
0
2
nn xx 21in oft coefficien
nnn xnn
xnn
xn
xnn
x 222
222
22
12
02
1
022110 oft coefficien
nnn
nnn
nnn
nnn
xn
nn
xn 2 oft coefficien
![Page 55: 12X1 T08 05 binomial coefficients (2010)](https://reader034.vdocuments.us/reader034/viewer/2022052412/5591a2211a28ab9a268b476b/html5/thumbnails/55.jpg)
Now nnn xxx 2111
![Page 56: 12X1 T08 05 binomial coefficients (2010)](https://reader034.vdocuments.us/reader034/viewer/2022052412/5591a2211a28ab9a268b476b/html5/thumbnails/56.jpg)
Now nnn xxx 2111
nn
knn
k
2
0
2
![Page 57: 12X1 T08 05 binomial coefficients (2010)](https://reader034.vdocuments.us/reader034/viewer/2022052412/5591a2211a28ab9a268b476b/html5/thumbnails/57.jpg)
Now nnn xxx 2111
nn
knn
k
2
0
2
!!!2
nnn
![Page 58: 12X1 T08 05 binomial coefficients (2010)](https://reader034.vdocuments.us/reader034/viewer/2022052412/5591a2211a28ab9a268b476b/html5/thumbnails/58.jpg)
Now nnn xxx 2111
nn
knn
k
2
0
2
!!!2
nnn
2!
!2nn
![Page 59: 12X1 T08 05 binomial coefficients (2010)](https://reader034.vdocuments.us/reader034/viewer/2022052412/5591a2211a28ab9a268b476b/html5/thumbnails/59.jpg)
Now nnn xxx 2111
nn
knn
k
2
0
2
!!!2
nnn
2!
!2nn
Exercise 5F; 4, 5, 6, 8, 10,15
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