Counting Principles NOTES Coach Bridges

The Fundamental Counting Principle If one event can occur in “m” ways and a second event can occur in “n” ways, the number of ways the two events can occur in sequence is m x n This rule can be extended for any number of events occurring in sequence.

Example of Counting Principle You are purchasing a new car. Using the following manufacturers, car sizes, and colors, how many different ways can you select one manufacturer, one car size and one color? Manufacturer: Ford, GM, Chrysler Car Size: Small, Medium Color: White (W), Red (R), Black (B), Green (G)

Answer 3 x 2 x 4 = 24 So there are 24 ways to select one manufacturer, one car size, and one color Now draw a Tree diagram using the information!

Permutations Is an important application of the Fundamental Counting Principle and determines the number of ways that “n” objects can be arranged in order An ordered arrangement of objects The number of different permutations of “n” distinct objects is n!.

Expression n! The expression “n!” is read as n factorial n! = n x (n – 1) x (n – 2) x (n – 3)…….. Ex. 0! = 1 1! = 1 2! = 2 x 1 = 2 3! = 3 x 2 x 1 = 6

Example The women’s hockey teams for the 2010 Olympics are Canada, Sweden, Switzerland, Slovakia, United States, Finland, Russia, and China. How many different final standings are possible?

Other Permutations Permutation of “n” objects taken “r” at a time Used when you want to choose some of the objects in a group and put them in order Distinguishable Permutations The number of ways “n” objects can occur, where you have many different types

“Order Matters” & Combinations Permutations are used when order matters Combination is used when order does not matter The number of ways to choose “r” objects from “n” objects without regard to order Refer to Table on Pg. 171