Section 3 4 Counting Principles Fundamental Counting Principle

Section 3. 4 Counting Principles

Fundamental Counting Principle If one event can occur m ways and a second event can occur n ways, the number of ways the two events can occur in sequence is m • n. This rule can be extended for any number of events occurring in a sequence. If a meal consists of 2 choices of soup, 3 main dishes and 2 desserts, how many different meals can be selected? Soup Dessert Main Start 2 • 3 • 2 = 12 meals

Factorials Suppose you want to arrange n objects in order. There are n choices for 1 st place. Leaving n – 1 choices for second, then n – 2 choices for third place and so on until there is one choice of last place. Using the Fundamental Counting Principle, the number of ways of arranging n objects is: n(n – 1)(n – 2)… 1 This is called n factorial and written as n!

Permutations A permutation is an ordered arrangement. The number of permutations for n objects is n! n! = n (n – 1) (n – 2)…. . 3 • 2 • 1 The number of permutations of n objects taken r at a time (where r £ n) is: You are required to read 5 books from a list of 8. In how many different orders can you do so? There are 6720 permutations of 8 books reading 5.

Combinations A combination is a selection of r objects from a group of n objects. The number of combinations of n objects taken r at a time is You are required to read 5 books from a list of 8. In how many different ways can you choose the books if order does not matter. There are 56 combinations of 8 objects taking 5.

1 2 3 Combinations of 4 objects choosing 2 1 3 1 4 2 3 4 4 Each of the 6 groups represents a combination. 4

1 2 4 3 Permutations of 4 objects choosing 2 1 2 3 1 1 2 3 3 4 1 1 4 4 2 2 4 4 Each of the 12 groups represents a permutation. 1 2 3 2

Homework: 1 -20 all pgs. 157 -158 Day 2: 21 -40 all pgs. 158 -159