Counting Principles A counting problem asks how many

Counting Principles

A counting problem asks “how many ways” some event can occur.

�To solve these counting problems, use the Fundamental Counting Principle. �Multiply the number of possibilities for each event.

Example #1 �You have 4 pairs of pants, 7 shirts, and 5 pairs of shoes. How many outfits are possible? ____ = 140

Example #2 A test contains 6 true/false questions, followed by 4 multiple choice questions with 5 possible answers. How many ways are there to answer all 10 questions? __ __ __ = 40000

Example #3 How many distinct license plates are possible if each must have 3 letters followed by 3 digits (0 -9)? ___ ___ ___ = 17, 576, 000

Example #4 �Suppose you are going to pick up 5 friends in a random order on your way to the mall. What is the probability that you pick your friends up in alphabetical order? �How many orders are possible? ____ = 120 ____

Cont. � So P(alphabetical order) =

Consider This… Suppose you have 2 extra tickets to an upcoming event. In deciding who to take, you have narrowed it down to your 4 best friends: Alex, Betty, Chris, and Diane. How many different groups of 2 can you choose?

Factorial The symbol for factorial is ! Examples: 4! = 4 x 3 x 2 x 1 = 24 7! = 7 x 6 x 5 x … x 2 x 1 = 5040 0! = 1 (By definition)

Combinations Combination – selecting r objects from a total of n objects where ORDER DOES NOT MATTER. Formula: C(n, r) = n! (n-r)! r! C(n, r) sometimes written as n. Cr

Combination Example – How many different committees of 5 students could be selected from a group of 20 students? C(20, 5) = 20! (20 -5)! 5! 15! 5! = 15504

Back to the earlier example You were counting the number of combinations of 2 students could be selected out of 4 C(4, 2) = 6 AB, AC, AD, BC, BD, CD Now consider, if the order mattered… AB AC AD BC BD CD BA CA DA CB DB DC Total of 12

Permutations Permutation – selecting r objects from a total of n objects where ORDER DOES MATTER. FORMULA: P(n, r) = n! (n-r)! P(n, r) sometimes written as n. Pr

Permutation Example � Example – How many ways a 1 st, 2 nd, and 3 rd place trophy could be awarded to a race of 20 students. � P(20, 3) = = 6840 20! (20 -3)! = 20! 17!

C(n, r) and P(n, r) on calc. Combinations, permutations, and factorials can all be done on the calculator. MATH PRB Assignment: 12. 1 Skip 1 -4 12. 2 Skip 20 and 24