1. Aim: What is the counting rule? Exam Tomorrow

2. Three Rules Sometimes we need to know all possible outcomes for a sequence of events – We use three rules 1. Fundamental counting rule 2. Permutation rule 3. Combination rule

3. Fundamental Counting Rule In a sequence of n events in which the first one has k 1 possibilities and the second event has k 2 and the third has k 3 and so forth, the total number of possibilities of the sequence will be k 1 * k 2 * k 3 … k n – In each case and means to multiply

4. Example A coin is tossed and a die is rolled. Find the number of outcomes for the sequences of events. – Solution: Since the coin can land either heads up or tails up and since the die can land with any one of six numbers face up, there are 2 * 6 = 12 possibilities A tree diagram can also be drawn for the sequences of events

5. Repetition When determining the number of possibilities of a sequence of events, one must know whether repetitions are permissible.

6. Example Example 1: The digits 0, 1, 2, 3, and 4 are to be used in a four-digit ID card. How many different cars are possible if repetitions are permitted? – Solution: 5 * 5 * 5 * 5 = 5 4 = 625 VS Example 2: The digits 0, 1, 2, 3, and 4 are to be used in a four-digit ID card. How many different cars are possible if repetitions are not permitted? – Solution: 5 * 4 * 3 * 2 = 120

7. Factorial Notation Uses exclamation point – 5! = 5 *4 * 3 * 2 * 1 For any counting n: n! = n(n-1)(n-2)(n-3)…1 0! = 1

8. Permutations Permutations: an arrangement of n objects in a specific order The arrangement of n objects in a specific order using r objects at a time is called a permutation of n objects taking r objects at atime – It is written as n P r and the formula is

9. Example There are three choices: A, B, C of which need to fill in 5 spots. Repetition is allowed. – Solution:

10. Combinations Combinations: a selection of distinct objects without regard to order The number of combinations of r objects selected from n objects is denoted by n C r and is given by the formula

11. Example How many combinations of 4 objects are there, taken 2 at a time? – Solution:

12. Exam Tomorrow Complete Review Sheet Solutions will be on website Study – In particular lessons 8 - 16