MAT 4830Numerical Analysis

Binomial Coefficients and Combinatorial Identities

http://myhome.spu.edu/lauw

Goals

Binomial Theorem Binomial Coefficients Combinatorial Identities Review shifting indices Review Induction

Take Home Exam

Need Binomial Coefficients for the second problem.

Need Binomial Theorem for a few parts of the second problem.

Binomial Expansion

2 2 2

3 3 2 2 3

2

3 3

? n

a b a ab b

a b a a b ab b

a b n Z

Binomial Theorem

0

!Binomial Coefficients:

,

,

! !

nn n r r

r

a b R

na b

n n

r

a br

n

Z

r r

n

Useful Formulas for Binomial Coefficients

, 0

1. 1 and 10

2.1

3.

Easy to check.

n Z r n

n n

n

nn

n n

n r r

Pascal’s Identity1

, 0 11 1

1, 1

1

n n nr n

r r r

OR

n n nr n

r r r

1, 0 1

1 1

1, 1

1

n n nr n

r r r

OR

n n nr n

r r r

Pascal’s Identity

Proof: Analysis

Binomial Theorem

0

, n

n n r r

r

na b a b n Z

r

Combinatorial Proof: Analysis

Example 1

Find the coefficient of in the expansion of .

Example 2

0

2 , n

n

r

nn Z

r

Proof: Analysis

0

,n

n n r r

r

na b a b

r

Example 3 (a)1

Given a fixed integer 0, , 1

n

i r

i nr n r

r r

Proof:1. Induction2. Can be done without induction, but need to take care special cases.

Analysis

1, 0 1

1 1

n n nr n

r r r

Example 3 (b)

1

Use (a) to find a formula for 1 2n

i

i n

Solution: Analysis

1

1

?

n

i r

i

r

r

n

r

Binomial Theorem

0

, n

n n r r

r

na b a b n Z

r

Induction Proof:Need some preparations

Analysis

Binomial Theorem

Proof: Analysis

11. , 0 1

1 1

2. 1 an

3. Index Shifting*

d 1, 0

n n nr n

r r r

m mm Z

m

0

, n

n n r r

r

na b a b n Z

r

3. Recall: Index Shifting for Summations (Use this if…)

1

1

1

1

( ) ( 1)

( 1)

n n

r m r m

n

r m

f r f r

f r

Index Shifting

Sigma representation of a summation is not unique

22226

2

22

22224

0

22

22225

1

22

543211

543211

54321

i

i

i

i

i

i

Index Shifting Rules

6

2

24

0

25

1

2 11iii

iii

Index Shifting Rules

6

2

24

0

25

1

2 11iii

iii

decrease the index by 1

increase the i in the summation by 1

Index Shifting Rules

6

2

24

0

25

1

2 11iii

iii

increase the index by 1

decrease the i in the summation by 1