Nonnegative Integer - Discrete Mathematics - Lecture Slides, Slides of Discrete Mathematics

Download Nonnegative Integer - Discrete Mathematics - Lecture Slides and more Slides Discrete Mathematics in PDF only on Docsity!

CS 173:

Discrete Mathematical Structures

Binomial Coefficients

(a + b) 2 = a 2 + 2ab + b^2

(a + b) 3 = a 3 + 3a 2 b + 3ab^2 + b^3

(a + b) 4 = a 4 + 4a 3 b + 6a 2 b^2 + 4ab^3

+ b^4 What is coefficient

of a^9 b^3 in (a + b) 12?

A. 36

B. 220

C. 15

D. 6

E. No clue

Binomial Coefficients

What is the coefficient of a 8 b 9 in the expansion of (3a +2b)^17?

Binomial Theorem: Let x and y be variables, and let

n be any nonnegative integer. Then

( x + y )

n

n

j

j = 0

n

∑^ x^

n − j

y

j

What is n? (^17)

What is j? (^9)

What is x? (^) 3a

What is y? (^) 2b

(3 a )^

8 (2 b ) 9 = 17

^3

8 2 9 a 8 b 9

Binomial Coefficients

(a + b) 4 = (a + b)(a + b)(a + b)(a + b)

= a^4

4 0

 



^ +^ a

413 b

 



^ +^ a

422 b 2

 



^ +^ ab^

(^433)  



^ +^ b^

(^444)  

 

A. 10 C 6

B. 9 C 4

C. 9 C 5

D. 8 C 4 + 8 C 5

E. No clue

CS 173

Binomial Coefficients

Sum each row of Pascal’s Triangle:

( x + y )^ n^ =

n j

j = 0

n ∑^ x^ n − j (^) y j

Powers of 2

Two proofs that

n

j

j = 0

n ∑ =^2 n

Let x=1 and y=1 in Binomial Theorem. Done

n

j

j = 0

n ∑^1

n − j 1 j = ( 1 + 1 ) n

n

j

j = 0

n ∑ =^2 n

Pascal’s Identity

A relationship between the entries in Pascal’s .

Suppose T is a set, |T|=n. Let a be an element in T, and let S = T - {a}. Let’s count the nC^ j subsets of size j.^ Note that some of these contain a, and some don’t. How many contain a? How many don’t?

n j

n-1 C^ j-

n - j -

 +^

n - j

n-1 C^ j

Combinations with repetition

Suppose you want to buy 5 bags of chips from

the 3 kinds you like at Meijer. In how many different ways can you stock up?

Out of 7 items, we are choosing 2 to be bars. From that, and our understanding of the model, we can report the answer.

7

2











 =^

7

5













Combinations with repetition

There are n+r-1Cr , r-sized combinations

from a set of n elements when

repetition is allowed.

Example: How many solutions are there to the equation

When the variables are nonnegative integers?

x 1 + x 2 + x 3 + x 4 = 10

13 C^3

Permutations with indistinguishable

objects

How many different strings can be made from the

letters in the phrase nano-nano?

Key thoughts: 8 positions, 3 kinds of letters to place.

In how many ways can we place the ns? 8 C^4 , now 4 spots are left

In how many ways can we place the as? 4 C^2 , now 2 spots are left

In how many ways can we place the os? 2 C^2 , now 0 spots are left

8

4













4

2













2

2











 =^

8! 4!4!

⋅

4! 2!2!

⋅

2! 2!0!

=

8! 4!2!2!

Permutations with indistinguishable

objects

How many distinct permutations are there of the

letters in the word APALACHICOLA?

12! 4! 2! 2!

How many if the two Ls must appear together?

11! 4! 2!

How many if the first letter must be an A?

11! 3!2! 2!

A little practice

A turtle begins at the upper left corner of a m x n

grid and meanders to the lower right corner.

How many routes could she take if she only moves

right and down, and if she must pass through the

dot at point (a,b)? a + b

a

 ^

  ⋅

m- a + n - bm- a  ^

 

A little practice

In how many ways can 11 identical computer science

books and 8 identical psychology books be

distributed among 5 students?

Hint: forget about the psychology books for the moment.

Hint: how can you combine your soln for the CS books with your soln for the Psych books?

15 11

 

  ⋅

(^128)  

 