Exploring Counting Rules & Probability

1. Aim: Find the number of possibilities. Exploring Counting Rules & Probability Counting

2. Aim: Find the number of possibilities. Try this … • Evaluate 6! • Evaluate = 6 x 5 x 4 x 3 x 2 x 1 OR = 720 8 x 7 x 6 x 5! = 8 x 7 x 6 = 336 = 5!

3. Aim: Find the number of possibilities. Fundamental Counting Principle • A soccer team has 3 different colored jerseys (red, white& gray), 2 different colored shorts (gray & black), and 1 colored socks (white). How many games can they play without repeating a uniform? . . 3 1 2 = 6 games jerseys shorts socks

4. Aim: Find the number of possibilities. Sometimes we need to consider the arrangement….

5. Aim: Find the number of possibilities. Factorial factorial (!) • When considering all pieces, use ___________. • When players of a basketball team are introduced at the beginning of a game, the public address announcer randomly chooses the order in which he reads off the names. If there are 12 players on the team, how many different orders are possible? • Suppose that the owner of the team insisted that the 5 starters be announced last, but said that it was fine to randomly select the order of the 7 reserves and the 5 starters. How many possible orders are there? 12! = 479,001,600 . 5! 7! = 604,800

6. Aim: Find the number of possibilities. Permutation does • Permutation: the arrangement _____ matter! OR • A football team has 25 different plays in their playbook and the coach randomly chooses 3 different plays to start the game. How many different sequences of plays do they have? • 25P3 • = 13,800 sequences

7. Aim: Find the number of possibilities. Combination does not • Combination: the arrangement _________ matter! OR • In track, meets with lots of competitors, sprinters often have to compete in several preliminary rounds to make it to the finals. In the first round, the competitors for each heat are chosen at random. Suppose that there are 32 sprinters in the 100-meter competition and that they will be randomly divided into 4 heats, so that there are 8 sprinters in each heat. How many different groups of sprinters can be chosen for the first heat? • 32C8 • = 10,518,300 groups

8. Aim: Find the number of possibilities. Permutation or Combination? Determine whether the following situation calls for using permutation or combination. • Select three students to attend a conference. • Elect a president, vice president, & secretary for student council. combination permutation

9. Aim: Find the number of possibilities. Sum it Up • The Fundamental Counting Principle says that if one event can occur in m ways and a second event can occur in n ways, then the first event and the second event can occur in _______ ways. • The factorial of the positive integer n, denoted n!, is the _________ of the integer n and all the positive integers below it. • Permutation are __________ sequences of objects in which each possible object occurs at most once, but not all possible objects need to be used. • Combination are _____________ collections of objects in which each possible object occurs at most once, but not all possible objects need to be used. m x n product ordered unordered