12x1 t08 05 binomial coefficients (2010)
TRANSCRIPT
Relationships Between Binomial Coefficients
Relationships Between Binomial Coefficients
Binomial Theorem
Relationships Between Binomial Coefficients
Binomial Theorem
n
k
kk
nn xCx0
1
Relationships Between Binomial Coefficients
Binomial Theorem
n
k
kk
nn xCx0
1
nn
nkk
nnnn xCxCxCxCC 2210
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)
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
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
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
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
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
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
7531b) CCCC nnnn
7531b) CCCC nnnn
n
k
kk
nn xCx0
1
7531b) CCCC nnnn
n
k
kk
nn xCx0
1
55
44
33
2210 xCxCxCxCxCC nnnnnn
7531b) CCCC nnnn
let x = 1;
n
k
kk
nn xCx0
1
55
44
33
2210 xCxCxCxCxCC nnnnnn
54321011 CCCCCC nnnnnnn
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
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
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
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)
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
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
n
kk
nCk1
c)
n
kk
nCk1
c)
n
k
kk
nn xCx0
1
n
kk
nCk1
c)
Differentiate both sides
n
k
kk
nn xCx0
1
n
kk
nCk1
c)
Differentiate both sides
n
k
kk
nn xCx0
1
1
0
11
kn
kk
nn xCkxn
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
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
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
n
k
knk
kC
0 11 d)
n
k
knk
kC
0 11 d)
n
k
kk
nn xCx0
1
n
k
knk
kC
0 11 d)
Integrate both sides
n
k
kk
nn xCx0
1
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
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
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
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
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
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
nnn
n
xxx
xii2111
identity; theof sidesboth on of tscoefficien theequatingBy
2
2
0 !!2
that;show
nn
kn
n
k
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
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
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
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
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
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
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
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
022110 oft coefficien
nnn
nnn
nnn
nnn
xn
kn
nkn
But
022110 oft coefficien
nnn
nnn
nnn
nnn
xn
kn
nkn
But 2222
210
nnnnn
022110 oft coefficien
nnn
nnn
nnn
nnn
xn
kn
nkn
But 2222
210
nnnnn
n
k kn
0
2
022110 oft coefficien
nnn
nnn
nnn
nnn
xn
kn
nkn
But 2222
210
nnnnn
n
k kn
0
2
nn xx 21in oft coefficien
022110 oft coefficien
nnn
nnn
nnn
nnn
xn
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
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
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
Now nnn xxx 2111
Now nnn xxx 2111
nn
knn
k
2
0
2
Now nnn xxx 2111
nn
knn
k
2
0
2
!!!2
nnn
Now nnn xxx 2111
nn
knn
k
2
0
2
!!!2
nnn
2!
!2nn
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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