Section 0.4 Day 1 – Counting Techniques EQ: How can I find the order and number of objects when order does and does not matter?

Probability Experiment Vocab Probability Experiment Outcome Sample Space Tree Diagram A trial process involving chance The result of a single trial of a process The set of all possible outcomes A way of listing all possible outcomes

Fundamental Counting Principle

2 different types of shoes – 9 different sizes Example 1 A shoe manufacturer makes brown and black shoes in nine different sizes. How many different shoes does the manufacturer make? 2 different types of shoes – 9 different sizes 2 * 9 = 18 different shoes

Factorial Counting Problems also involve determining the number of different objects in a certain order. This is called a permutation. The number of permutations of n distinct objects is n!

This is also knows as a factorial. In this case it would be 9! Example 2 There are 9 players in a baseball lineup. No two players can share the same spot in the lineup. How many lineups are possible? 9 players in a line up __ __ __ __ __ __ __ __ __ 9 8 7 6 5 4 3 2 1 This is also knows as a factorial. In this case it would be 9! First spot in the lineup there are 9 spots to choose from Since there can be no repeats in positions the second spot in the lineup there are now only 8 spots to choose from This continues all the way till the last spot in the lineup

Permutation An arrangement of a group of distinct objects in a certain order. *Order matters*

How to put combination and permutations into the calculator Example 3 Student council is selecting a President, Vice President, and Secretary from a group of 15 students. How many ways can they select these positions? How to put combination and permutations into the calculator Order matters of picking the President, Vice President, and Secretary. You are picking 3 objects (r) from a group of 15 (n) P(15, 3) = 2, 730 different ways to select these positions

Combination A selection of distinct objects in which the order of the objects is NOT important.

Example 4 A restaurant offers a total of 8 side dishes. How many different ways can a customer choose 3 side dishes? Combination because the order that you choose the side dishes doesn’t matter. C (8,3) = 56 different ways to choose side dishes

Example 5 300 people buy tickets for a raffle. Then 5 different raffle tickets are drawn at random to claim prizes. If there are 5 different prizes, will this be a combination or a permutation? If there is only 1 type of prize? Permutation – order that you hand out those different prizes matters Combination – order that you hand out the prizes doesn’t matter because they are all the same prize