General Expansion of Binomials

General Expansion of Binomials

kkk

n xxC 1in oft coefficien theis

General Expansion of Binomials

kkk

n xxC 1in oft coefficien theis

k

nCkn

General Expansion of Binomials

kkk

n xxC 1in oft coefficien theis

nn

nnnnn xCxCxCCx 22101

k

nCkn

General Expansion of Binomials

kkk

n xxC 1in oft coefficien theis

nn

nnnnn xCxCxCCx 22101

which extends to;

nn

nnn

nnnnnnnn bCabCbaCbaCaCba

11

222

110

k

nCkn

General Expansion of Binomials

kkk

n xxC 1in oft coefficien theis

nn

nnnnn xCxCxCCx 22101

432.. xge

which extends to;

nn

nnn

nnnnnnnn bCabCbaCbaCaCba

11

222

110

k

nCkn

General Expansion of Binomials

kkk

n xxC 1in oft coefficien theis

nn

nnnnn xCxCxCCx 22101

432.. xge

which extends to;

nn

nnn

nnnnnnnn bCabCbaCbaCaCba

11

222

110

44

433

4222

431

440

4 33232322 xCxCxCxCC

k

nCkn

General Expansion of Binomials

kkk

n xxC 1in oft coefficien theis

nn

nnnnn xCxCxCCx 22101

432.. xge

which extends to;

nn

nnn

nnnnnnnn bCabCbaCbaCaCba

11

222

110

44

433

4222

431

440

4 33232322 xCxCxCxCC 432 812162169616 xxxx

k

nCkn

Pascal’s Triangle Relationships

Pascal’s Triangle Relationships

11 where 1 11

1

nkCCC kn

kn

kn

Pascal’s Triangle Relationships

11 where 1 11

1

nkCCC kn

kn

kn

1111 nn xxx

Pascal’s Triangle Relationships

11 where 1 11

1

nkCCC kn

kn

kn

1111 nn xxx

11

1111

11

10

11

nn

nkk

nkk

nnn xCxCxCxCCx

Pascal’s Triangle Relationships

11 where 1 11

1

nkCCC kn

kn

kn

1111 nn xxx

11

1111

11

10

11

nn

nkk

nkk

nnn xCxCxCxCCx

kx of tscoefficienat looking

Pascal’s Triangle Relationships

11 where 1 11

1

nkCCC kn

kn

kn

1111 nn xxx

11

1111

11

10

11

nn

nkk

nkk

nnn xCxCxCxCCx

kx of tscoefficienat looking

knCLHS

Pascal’s Triangle Relationships

11 where 1 11

1

nkCCC kn

kn

kn

1111 nn xxx

11

1111

11

10

11

nn

nkk

nkk

nnn xCxCxCxCCx

kx of tscoefficienat looking

knCLHS

Pascal’s Triangle Relationships

11 where 1 11

1

nkCCC kn

kn

kn

1111 nn xxx

11

1111

11

10

11

nn

nkk

nkk

nnn xCxCxCxCCx

kx of tscoefficienat looking

knCLHS kn

kn CCRHS 1

11 11

Pascal’s Triangle Relationships

11 where 1 11

1

nkCCC kn

kn

kn

1111 nn xxx

11

1111

11

10

11

nn

nkk

nkk

nnn xCxCxCxCCx

kx of tscoefficienat looking

knCLHS kn

kn CCRHS 1

11 11

kn

kn CC 1

11

Pascal’s Triangle Relationships

11 where 1 11

1

nkCCC kn

kn

kn

1111 nn xxx

11

1111

11

10

11

nn

nkk

nkk

nnn xCxCxCxCCx

kx of tscoefficienat looking

knCLHS kn

kn CCRHS 1

11 11

kn

kn CC 1

11

k

nk

nk

n CCC 11

1

Pascal’s Triangle Relationships

11 where 1 11

1

nkCCC kn

kn

kn

1111 nn xxx

11

1111

11

10

11

nn

nkk

nkk

nnn xCxCxCxCCx

kx of tscoefficienat looking

knCLHS kn

kn CCRHS 1

11 11

kn

kn CC 1

11

k

nk

nk

n CCC 11

1

l"symmetrica is trianglesPascal'"

11 where 2 nkCC knn

kn

Pascal’s Triangle Relationships

11 where 1 11

1

nkCCC kn

kn

kn

1111 nn xxx

11

1111

11

10

11

nn

nkk

nkk

nnn xCxCxCxCCx

kx of tscoefficienat looking

knCLHS kn

kn CCRHS 1

11 11

kn

kn CC 1

11

k

nk

nk

n CCC 11

1

l"symmetrica is trianglesPascal'"

11 where 2 nkCC knn

kn

1 3 0 nnn CC

Pascal’s Triangle Relationships

11 where 1 11

1

nkCCC kn

kn

kn

1111 nn xxx

11

1111

11

10

11

nn

nkk

nkk

nnn xCxCxCxCCx

kx of tscoefficienat looking

knCLHS kn

kn CCRHS 1

11 11

kn

kn CC 1

11

k

nk

nk

n CCC 11

1

l"symmetrica is trianglesPascal'"

11 where 2 nkCC knn

kn

1 3 0 nnn CC

Exercise 5B; 2ace, 5, 6ac,10ac, 11, 14