Fundamental Counting Principle

1. 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? Dessert Soup Main Start • • 2 3 2 = 12 meals

2. Factorials Suppose you want to arrange n objects in order. There are n choices for 1st 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!

3. 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 takenr 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.

4. Combinations A combination is a selection ofr objects from a group ofn objects. The number of combinations of n objects taken rat 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.

5. 2 1 3 1 2 4 1 3 Combinations of 4 objects choosing 2 4 1 2 3 4 3 2 4 Each of the 6 groups represents a combination.

6. 2 2 1 1 3 3 1 1 2 4 1 3 Permutations of 4 objects choosing 2 4 4 1 1 2 2 3 3 4 4 3 3 2 2 4 4 Each of the 12 groups represents a permutation.