Warm Up 1/31/11 1. If you were to throw a dart at the purple area, what would be the probability of hitting it? I I 5

13 20 I I 5 Area of Blue = 20·13 A = 260 Area of purple = ½ (8)(3) A = 12 3 Probability of hitting purple = 12/260 P = 3/65

15.2 Objective: To use the FUNDAMENTAL COUNTING PRINCIPLE and PERMUTATIONS to find the possible number of arrangements

Remember… 6! = 6·5·4·3·2·1 = 720 This is called a FACTORIAL Examples: 1) 5! = 5·4·3·2·1 = 120 2) = 840

A. Arrangements Example 1: How many possible 3-letter arrangements can be made using the 26 letters of the alphabet? (repetition is allowed) ___ ___ ___ We can find the total number by multiplying all 3 together… 26·26·26 = 17, 576 This is called the FUNDAMENTAL COUNTING PRINCIPLE, which allows us to multiply together the possible outcomes for a series of events.

Example 2: How many 7-digit phone numbers can be created using 0-9? (Restriction: the first 2 #’s can NOT be 0 or 1) ___ ___ ___ ___ ___ ___ ___ Total possibilities = 6,400,000

B. Permutations There is a special type of arrangement called a PERMUTATION: *repetition IS NOT allowed *the order is important Example 1: How many 4-letter permutations can be made using the letters A, B, C and D? ___ ___

Example 2: Brad is creating a 7 character screen name. The first 3 characters must be a letter from his name, and the last 4 characters must be a digit from the year How many different permutations are there? ___ ___ ___ ___ ___ ___ ___

Example 3: How many 5-letter permutations can be made using the letters in the word “FISHER”? ___ ___ ___ ___ ___ Another way this can be written is: Total # of items The # we want In General:

Calculate the following: 1) 2)3) 20120

C. Permutations with repeating letters: If there are repeating letters in a word with n total letters, to find the number of permutations we use: Where represent the number of times that a letter repeats itself.

Example 1 How many 7-letter permutations can be made from the letters in the word “CLASSIC” ?

Example 2 How many 11-letter permutations can be made from the letters in MISSISSIPPI?