02.25.09

Discrete Mathematics

Exercises

Exercise 1: There are 18 Computer Science (CS) majors and 325

Business Administration (BA) majors at a college How many ways are there to pick two representatives, so

that one is a CS major and the other is a BA major? How many ways are there to pick one representative who

is either a CS major or a BA major?

Exercises

Exercise 2: How many different three-letter initials can people

have? How many different three-letter initials with none of

the letters repeated can people have? How many different three-letter initials are there that

begin with an A?

Exercises

Exercise 3: How many positive integers less than 1000

a) are divisible by 7?b) are divisible by 7 but not by 11?c) are divisible by both 7 and 11?d) are divisible by either 7 or 11?

Exercises

Exercise 3: How many positive integers less than 1000

a) are divisible by 7?b) are divisible by 7 but not by 11?c) are divisible by both 7 and 11?d) are divisible by either 7 or 11?

Exercises

Exercise 4: Suppose that there are 1807 freshmen at a college. Of

these, 453 are taking a course in Computer Science (CS), 567 are taking a course in Business Administration (BA), and 299 are taking courses in both CS and BA. How many are not taking a course either in CS or BA?

Today’s Topics

Pigeonhole PrinciplePermutation

Pigeonhole Principle

The Pigeonhole Principle states that, given two natural numbers n and m with n > m, if n items are put into m pigeonholes, then at least one pigeonhole must contain more than one item.

Pigeonhole Principle

Examples: Among any group of 367 people, there must be at

least two with the same birthday, because there are only 366 possible birthdays.

In any group of 27 English words, there must be at least two that begin with the same letter, since there are 26 letters in the English alphabet.

Pigeonhole Principle

Exercise 1: How many students must be in a class to guarantee

that at least two students receive the same score on the final exam, if the exam is graded on a scale from 0 to 100 points?

Pigeonhole Principle

Exercise 2: Assume that in a box there are 10 black socks and 12

blue socks and you need to get one pair of socks of the same color. Supposing you can take socks out of the box only once and only without looking, what is the minimum number of socks you'd have to pull out at the same time in order to guarantee a pair of the same color?

Pigeonhole Principle

The Generalized Pigeonhole Principle states that if N objects are placed into k boxes, then there is at least one box containing at least [N/k] objects.

Example: Among 100 people there are at least [100/12] = 9 who

were born in the same month.

Pigeonhole Principle

Exercise 3: What is the minimum number of students required in

a discrete math class to be sure that at least six will receive the same grade, if there are 5 possible grades, A, B, C, D, and F?

Pigeonhole Principle

Exercise 4: How many cards must be selected from a standard

deck of 52 cards to guarantee that at least three cards of the same suit are chosen?

Permutation

Permutation An ordered arrangement of the elements of a set

r-Permutation An ordered arrangement of r elements of a set

P(n,r) The number of r-permutations of a set with n elements

Permutation

Example 1: How many ways are there to select a first-prize

winner, second-prize winner, and a third-prize winner from 100 different people who have entered a contest?

Example 2: Let S = {1, 2, 3, 4, 5} What is the 3-permutations of S?