Chapter 1: Introduction Math 4203

1. Spring 2011 AyonaChatterjee Chapter 1: IntroductionMath 4203

2. Theorem1: Basic principle of counting • If an operation consists of two steps, of which the first can be done in n1 ways and the second can be done in n2 ways, then the whole operation can be done in n1 n2 ways.

3. Example • How many different possible sets of initials are there for people with a first, middle and last name? • For the first name initial there are 26 possibilities. • For the middle name initial there are another 26 possibilities • For the last name initial there are 26 possibilities.(You can have the same initial for your first , middle and last name) • Total possible sets of initials = 26 x 26 x 26 = 17576

4. Permutations • An arrangement of items such that • r items are chosen at a time from n distinct items. • Repetition of an item is not allowed. • The order of an item is important. • Notation

5. Example • Suppose a group of four women would like to do Secret Santa this holiday. Calculate how many possible different permutations of gift buying there are for the four women. • Note- Secret Santa is a method whereby each member of a group anonymously buys a holiday gift for another member of the group. Each person is secretly assigned to buy a gift for another randomly chosen person in the group. • There are 4 women and two people are associated with a gift. Each person can receive and give only one gift, so repetition is not allowed. Also order is important as A giving gift to B is different from B giving gift to A.

6. Theorems for Permutations • The number of permutations of n distinct objects arranged in a circle is (n-1)!. This is called circular permutations. • The number of permutations of n objects of which n1 are of one kind, n2 are of a second kind, …., nk are of the kth kind and n1 + n2 + … nk =n is

7. Examples • How many circular permutations are there for 8 people playing bridge? • Here we have(8-1)!= 7! • How many different permutations are there of the letters in the word “book”? • Here we have 4!/2! = 12.

8. Combinations • An arrangement of items in which • r items are chosen from n distinct items • Repetition of items is not allowed. • Order of the items is not important. • Notation

9. Theorem • The number of ways in which a set of n distinct objects can be partitioned into k subsets with n1 objects in the first subset, n2 objects in the second subset, and nk objects in the kth subset is

10. Examples • In how many ways can we select a committee of 5 from a group of 7? • In how many ways can seven businessmen attending a convention be assigned one triple and two double hotel rooms? • Here n = 7, n1 = 3, n2 = 2, n3 = 2 we get

11. Binomial Coefficients

12. Theorems • Theorem: For any positive integers n and r