12x1 t08 05 binomial coefficients (2011)

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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