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Chapter 5 The Binomial Coefficients

Chapter 5 The Binomial Coefficients. Summary Pascal’s formula The binomial theorem Identities Unimodality of binomial coefficients The multinomial theorem

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Page 1: Chapter 5 The Binomial Coefficients. Summary Pascal’s formula The binomial theorem Identities Unimodality of binomial coefficients The multinomial theorem

Chapter 5

The Binomial Coefficients

Page 2: Chapter 5 The Binomial Coefficients. Summary Pascal’s formula The binomial theorem Identities Unimodality of binomial coefficients The multinomial theorem

Summary

• Pascal’s formula

• The binomial theorem

• Identities

• Unimodality of binomial coefficients

• The multinomial theorem

• Newton’s binomial theorem

Page 3: Chapter 5 The Binomial Coefficients. Summary Pascal’s formula The binomial theorem Identities Unimodality of binomial coefficients The multinomial theorem

Review

• If k > n, C(n,k) = 0, C(n, 0) =1; If n is positive and 1≤k ≤n, then

• C(n,r) = C(n, n−r)

)!(!

!

!

)!/(!

),(

),(),(

rnr

n

r

rnn

rrP

rnP

r

nrnC

Page 4: Chapter 5 The Binomial Coefficients. Summary Pascal’s formula The binomial theorem Identities Unimodality of binomial coefficients The multinomial theorem

Pascal’s formula

• For all integers n and k with 1≤k ≤n-1,

• Hint: Let S be a set of n elements. We distinguish one of the elements of S and denote it by x. We then partition the set X of k-combinations of S into two parts , A and B such that all those k-combinations in A do not contain x while those in B contain x. Then

C(n, k) = |A| + |B| = C(n-1, k) + C(n-1, k-1).

1

11

k

n

k

n

k

n

Page 5: Chapter 5 The Binomial Coefficients. Summary Pascal’s formula The binomial theorem Identities Unimodality of binomial coefficients The multinomial theorem

Binomial Theorem

Let n be a positive integer. Then for all x and y,

In summation notation,

nnnnnn yyxn

nyx

nyx

nxyx

11221

121)(

n

k

kknn yxk

nyx

0

)(

Page 6: Chapter 5 The Binomial Coefficients. Summary Pascal’s formula The binomial theorem Identities Unimodality of binomial coefficients The multinomial theorem

Exercises

• Expand (x+y)5 and (x+y)6, using the binomial theorem.

• Expand (2x-y)7, using the binomial theorem.

Page 7: Chapter 5 The Binomial Coefficients. Summary Pascal’s formula The binomial theorem Identities Unimodality of binomial coefficients The multinomial theorem

Equivalent forms

n

k

knkn

n

k

knkn

n

k

kknn

yxk

nyx

yxkn

nyx

yxkn

nyx

0

0

0

)(

,)(

,)(

Page 8: Chapter 5 The Binomial Coefficients. Summary Pascal’s formula The binomial theorem Identities Unimodality of binomial coefficients The multinomial theorem

Special case

Let n be a positive number. Then for all x,

kn

k

kn

k

n xkn

nx

k

nx

00

1

Page 9: Chapter 5 The Binomial Coefficients. Summary Pascal’s formula The binomial theorem Identities Unimodality of binomial coefficients The multinomial theorem

Identities

• For positive integers n and k,

).0(,2

,22

21

1

,23120

,1

1

0

2

1

1

nn

n

k

n

nn

nn

nn

nnnn

k

nn

k

nk

n

k

n

n

Page 10: Chapter 5 The Binomial Coefficients. Summary Pascal’s formula The binomial theorem Identities Unimodality of binomial coefficients The multinomial theorem

Hints for proof

• Trivial ;• Set x=1 and y=-1 in the binomial theorem;• Differentiate both sides with respect to x for the

special case of binomial theorem and then substituting x=1;

• Counts the number of n-combinations of S (a set with 2n elements). Partition S into two subsets A and B. Each n-combination of S contains k elements of A and the remaining n-k elements in B. Note that C(n, k) = C(n, n-k).

Page 11: Chapter 5 The Binomial Coefficients. Summary Pascal’s formula The binomial theorem Identities Unimodality of binomial coefficients The multinomial theorem

Exercises

• Use the binomial theorem to prove that

• Generalize to find the sum for any real number r.

• Vandermonde convolution: for all positive integers mi, m2 and n,

.230

n

k

kn

k

n

n

k

krk

n

0

.0

2121

n

k n

mm

kn

m

k

m

Page 12: Chapter 5 The Binomial Coefficients. Summary Pascal’s formula The binomial theorem Identities Unimodality of binomial coefficients The multinomial theorem

Generalization

• Let r be any real number and k be any integer (positive, negative, or zero).

).1(

)0(

)1(

0

1!

)1()1(

kif

kif

kifk

krrr

k

r

Page 13: Chapter 5 The Binomial Coefficients. Summary Pascal’s formula The binomial theorem Identities Unimodality of binomial coefficients The multinomial theorem

Identities

For any real number r and integer k,

.1

1110

0

,1

1

1

0

k

n

k

n

k

n

kk

kfor

k

kr

k

krrr

Page 14: Chapter 5 The Binomial Coefficients. Summary Pascal’s formula The binomial theorem Identities Unimodality of binomial coefficients The multinomial theorem

Unimodality of binomial coefficients

Let n be a positive integer. The sequence of binomial coefficients is a unimodal sequence. More precisely, if n is even,

and if n is odd,

,1/210

n

n

n

n

n

nnn

.12/12/110

n

n

n

n

n

n

n

nnn

Page 15: Chapter 5 The Binomial Coefficients. Summary Pascal’s formula The binomial theorem Identities Unimodality of binomial coefficients The multinomial theorem

A corollary

• For n a positive integer, the largest of the binomial coefficients

.

n/2

n

n/2

n

n

n,,

2

n,

1

n,

0

n

is

Page 16: Chapter 5 The Binomial Coefficients. Summary Pascal’s formula The binomial theorem Identities Unimodality of binomial coefficients The multinomial theorem

Clutter

• Let S be a set of n elements. A Clutter of S is a collection C of combinations of S with the property that no combination in C is contained in another.

• Example: if S ={a, b, c, d} then

C = {{a, b}, {b, c, d}, {a, d} , {a, c}} is a clutter.

Page 17: Chapter 5 The Binomial Coefficients. Summary Pascal’s formula The binomial theorem Identities Unimodality of binomial coefficients The multinomial theorem

Sperner’s theorem

• Let S be a set of n elements. Then a clutter on S contains at most sets.

2/n

n

Page 18: Chapter 5 The Binomial Coefficients. Summary Pascal’s formula The binomial theorem Identities Unimodality of binomial coefficients The multinomial theorem

Multinomial coefficients

here n1,n2, …nt are non-nagative integers with n1+n2+ …+nt = n.

!!!

!

2121 tt nnn

n

nnn

n

Page 19: Chapter 5 The Binomial Coefficients. Summary Pascal’s formula The binomial theorem Identities Unimodality of binomial coefficients The multinomial theorem

Pascal’s formula for the multinomial coefficients

• Pascal’s formula for binomial coefficients:

• Pascal’s formula for multinomial coefficients

knk

n

knk

n

knk

n

k

n

k

n

k

n

1

1

1

1

1

11

.1

1

1

1

1

1

2121

2121

tt

tt

nnn

n

nnn

n

nnn

n

nnn

n

Page 20: Chapter 5 The Binomial Coefficients. Summary Pascal’s formula The binomial theorem Identities Unimodality of binomial coefficients The multinomial theorem

The multinomial theorem

Let n be a positive integer. For all x1, x2, …,xt,

where the summation extends over all non-negative integral solutions x1, x2, …,xt of x1+ x2+ …+xt = n.

tnt

nn

t

nt xxx

nnn

nxxx

21

2121

21 )(

Page 21: Chapter 5 The Binomial Coefficients. Summary Pascal’s formula The binomial theorem Identities Unimodality of binomial coefficients The multinomial theorem

Example and exercise

• When (x1+ x2+ …+x5)7 is expanded, the coefficient of x1

2x3x43x5 equals

• When (2x1 3x﹣ 2+5x3)6 is expanded, what the coefficient of x1

3x2x32 is?

.420!1!3!1!0!2

!7

13102

7

Page 22: Chapter 5 The Binomial Coefficients. Summary Pascal’s formula The binomial theorem Identities Unimodality of binomial coefficients The multinomial theorem

Newton’s Binomial Theorem

Let a be a real number . Then for all x and y with 0 ≤ |x| <|y|,

where

0

)(k

kkaa yxk

ayx

!

121

k

kaaaa

k

a

Page 23: Chapter 5 The Binomial Coefficients. Summary Pascal’s formula The binomial theorem Identities Unimodality of binomial coefficients The multinomial theorem

Special case

For any real number a,

k

k

a xk

ax

0

1

Page 24: Chapter 5 The Binomial Coefficients. Summary Pascal’s formula The binomial theorem Identities Unimodality of binomial coefficients The multinomial theorem

Correspondence

e.g. If a is a positive integer n, then when k>n0

k

n

kn

k

n xk

nx

0

1So

Page 25: Chapter 5 The Binomial Coefficients. Summary Pascal’s formula The binomial theorem Identities Unimodality of binomial coefficients The multinomial theorem

When a = -n

We can verify that

!k

knnn

k

n 11

!k

knnnk 111

1

11

11

n

kn

k

kn kk

Page 26: Chapter 5 The Binomial Coefficients. Summary Pascal’s formula The binomial theorem Identities Unimodality of binomial coefficients The multinomial theorem

For |y|<1

0 1

111

k

kkn yn

kny

Set xy

0 1

111

k

kkn xn

knx

0

11

1

k

kxn

kn

0 1

1

k

kxn

kn

Page 27: Chapter 5 The Binomial Coefficients. Summary Pascal’s formula The binomial theorem Identities Unimodality of binomial coefficients The multinomial theorem

For |y| < 1 and let n=1

0

1

0

1

1

11

k

k

k

kk

yy

yy

Page 28: Chapter 5 The Binomial Coefficients. Summary Pascal’s formula The binomial theorem Identities Unimodality of binomial coefficients The multinomial theorem

For a = 1/2

!k

k

k

1

21

321

221

121

21

21

!k

k

232

25

23

21

21

!k

kk

k

2325311 1

Page 29: Chapter 5 The Binomial Coefficients. Summary Pascal’s formula The binomial theorem Identities Unimodality of binomial coefficients The multinomial theorem

For a = 1/2 (cont’d)

!k

k

k k

k

2325311 1

21

!kk

kkk

k

2226422232543211 1

!

!

kk

kkk

k

213212221

1

1

1

22

21

12221

12

1

12

1

k

k

kkk

kk

k

k

k

!!

!

Page 30: Chapter 5 The Binomial Coefficients. Summary Pascal’s formula The binomial theorem Identities Unimodality of binomial coefficients The multinomial theorem

Special case for a =1/2

Consequently

k

kk

kk

k

xk

k

kx

kx

012

1

0

2

1

1

22

2

11 2

1

Page 31: Chapter 5 The Binomial Coefficients. Summary Pascal’s formula The binomial theorem Identities Unimodality of binomial coefficients The multinomial theorem

Assignments

• Exercises 6, 8, 11,15, 22, 34, 36 and 42 in the text book.