Fall 2014 COMP 2300 Discrete Structures for Computation Donghyun (David) Kim Department of Mathematics and Computer Science North Carolina Central University

1

Fall 2014COMP 2300 Discrete Structures for Computation

Donghyun (David) KimDepartment of Mathematics and Computer ScienceNorth Carolina Central University

Chapter 9.7Pascal’s Formula and the Binomial Theorem

2

Pascal’s Formula•

• Pascal’s formula: Let n and r be positive inte-gers and suppose . Then,

why?

Fall 2014 COMP 2300 Department of Mathematics and Computer Science Donghyun (David) Kim North Carolina Central University

rn

n

rrn

n

r

n

!)!(!

r

n

r

n

r

n 1

1

nr

3

Pascal’s Formula – cont’•

• Pascal’s formula: Let n and r be positive inte-gers and suppose . Then,


rn

n

rrn

n

r

n

!)!(!

r

n

rrn

n

rrn

nn

rrn

rnrn

rrn

rnn

rrn

rn

rrn

n

rrn

n

r

n

r

n

1

!)!1()!1(

!))!1(()1!(

!))!1(()1!(

!))!1(())1(!(

!))!1((!

!)!(!

)!1))!(1((!

1

nr

4

Pascal’sTriangle


0

0

0

1

1

2

2

4

1

3

1

3

r

n

r

n

r

n 1

1

5

Deriving New Formulas from Pascal’s Formula• Use Pascal’s formula to derive a formula for

in

terms of values of , , and . Assume n and

r are nonnegative integers and .


r

n 2

r

n

1r

n

2r

n

nr 2

6

Deriving New Formulas from Pascal’s Formula – cont’• Use Pascal’s formula to derive a formula for

in

terms of values of , , and . Assume n and

r are nonnegative integers and .


r

n 2

r

n

1r

n

2r

n

nr 2

r

n

r

n

r

n

r

n

r

n

r

n

r

n

r

n

r

n

r

n

r

n

r

n

1

1

121

1

12

2

1

1

12

7

The Binomial Theorem• In algebra a sum of two terms, such

as , is called a binomial.

• The binomial theorem gives an expression for the powers of a binomial , for each positive integer n and all real numbers a and b.


ba

nba )(

8

The Binomial Theorem – cont’•


4

4

3

4

2

4

1

4

0

4)(

3

3

2

3

1

3

0

3)(

2

2

1

2

0

2)(

43122344

32233

222

bbababaaba

babbaaba

bababa

ba

nnnnnn

k

kknn bban

nba

nba

naba

k

nba

112211

0 121)(

9

The Binomial Theorem – cont’• Proof of the Binomial Theorem (By Math. In-

duction)

• When n=1,

• Suppose

Then, we have


nnnnnn

k

kknn bban

nba

nba

naba

k

nba

112211

0 121)(

babababak

bak

kk

10011

0

11 1)(

n

k

kknn bak

nba

0

)(

n

k

kknnn bak

nbabababa

0

1 )())(()(

10

Deriving Another Combinatorial Identity from the Binomial Theorem• Use the binomial theorem to show that

for all integers


n

nnnn

k

nn

k

n 210

20

.0n

11

Deriving Another Combinatorial Identity from the Binomial Theorem• Use the binomial theorem to show that

for all integers


n

nnnn

k

nn

k

n 210

20

.0n

kknn

k

n

k

n

k

n

k

n112

00

n

nnnn nnnn 112

111

110

11 022110

12

Using a Combinatorial Argument to Derive the Identity• Show that


n

n

nnnn2

210

13

Using a Combinatorial Argument to Derive the Identity – cont’• Show that

• Suppose S is a set with n elements.


n

n

nnnn2

210

Number of subsets of

S


size 0


size 1


size n

n2

14

Using a Combinatorial Argument to Derive the Identity – cont’• Show that

• Suppose S is a set with n elements.


n

n

nnnn2

210


S


size 0


size 1


size n

0

n

1

n

n

nn2

k

n

15

Using the Binomial Theorem to Sim-plify a Sum• Express the following sum in closed form

(without using a summation symbol and without using an ellipsis…):


n

k

k

k

n

0

9

16

Using the Binomial Theorem to Sim-plify a Sum – cont’• Express the following sum in closed form

(without using a summation symbol and without using an ellipsis…):

•


n

k

k

k

n

0

9

nnn

k

kknn

k

k

k

n

k

n1091919

00

)(

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Fall 2014 COMP 2300 Discrete Structures for Computation Donghyun (David) Kim Department of Mathematics and Computer Science North Carolina Central University