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CS 173: Discrete Mathematical Structures

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CS 173:Discrete Mathematical Structures

Cinda Heeren

Siebel Center, rm 2213

Office Hours: W 12:30-2:30

- Homework #7 due 10/23, 8a.
- Exam #2, 11/3, 7-9p.
- No class 11/3.

Cs173 - Spring 2004

Why am I showing this to you?

- Beautiful patterns
- Recursive defn
- New type of proof
- Applications in more complex counting techniques

(a + b)2 = a2 + 2ab + b2

(a + b)3 = a3 + 3ab2 + 3a2b + b3

(a + b)4 = a4 + 4ab3 + 6a2b2 + 4a3b + b4

Cs173 - Spring 2004

What is coefficient of a9b3 in (a + b)12?

- 36
- 220
- 15
- 6
- No clue

(a + b)2 = a2 + 2ab + b2

(a + b)3 = a3 + 3a2b + 3ab2 + b3

(a + b)4 = a4 + 4a3b + 6a2b2 + 4ab3 + b4

Cs173 - Spring 2004

= a4

+ a3b

+ a2b2

+ ab3

+ b4

Binomial Theorem: Let x and y be variables, and let n be any nonnegative integer. Then

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

Cs173 - Spring 2004

Binomial Theorem: Let x and y be variables, and let n be any nonnegative integer. Then

17

9

3a

2b

What is the coefficient of a8b9 in the expansion of (3a +2b)17?

What is n?

What is j?

What is x?

Cs173 - Spring 2004

What is y?

= a4

+ a3b

+ a2b2

+ ab3

+ b4

- 10C6
- 9C4
- 9C5
- 8C4 + 8C5
- No clue

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

Cs173 - Spring 2004

Powers of 2

2n

nC0

nC1

nC2

Sum each row of Pascal’s Triangle:

Two proofs that

Suppose you have a set of size n. How many subsets does it have?

How many subsets of size 0 does it have?

How many subsets of size 1 does it have?

How many subsets of size 2 does it have?

Cs173 - Spring 2004

Count all subsets in this way, and we have the result!

Powers of 2

Done

Sum each row of Pascal’s Triangle:

Two proofs that

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

Cs173 - Spring 2004

n-1Cj-1

n-1Cj

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 nCj subsets of size j. Note that some of these contain a, and some don’t.

How many contain a?

How many don’t?

Cs173 - Spring 2004

A

m items

B

n items

Let m, n, and r be nonnegative integers with r not exceeding either m or n. Then

To choose r items, take some from A and some from B. All possible ways of doing this gives the result.

Cs173 - Spring 2004

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.

Cs173 - Spring 2004

Example: How many solutions are there to the equation

When the variables are nonnegative integers?

13C3

There are n+r-1Cr, r-sized combinations from a set of n elements when repetition is allowed.

Cs173 - Spring 2004

6

3

How many different strings can be made from the letters in the word rat?

How many different strings can be made from the letters in the word egg?

Cs173 - Spring 2004

8C4, now 4 spots are left

4C2, now 2 spots are left

2C2, now 0 spots are left

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?

In how many ways can we place the as?

In how many ways can we place the os?

Cs173 - Spring 2004

How many distinct permutations are there of the letters in the word APALACHICOLA?

How many if the two Ls must appear together?

How many if the first letter must be an A?

Cs173 - Spring 2004

A turtle begins at the upper left corner of an n x m grid and meanders to the lower right corner.

How many routes could she take if she only moves right and down?

Cs173 - Spring 2004

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?

Cs173 - Spring 2004