Lesson A7 - Counting Techniques and Permutations. Learning Goals:

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1 Learning Goals: * Determine tools and strategies that will determine outcomes more efficiently * Use factorial notation effectively * Determine probabilities for simple ordered events

2 Example 1: You are trying to put three children - represented by A, B, and C - in a line for a game. a) How many different orders are possible? b) What is the probability that a random ordering will produce the order ABC? Solution: a) Construct a tree diagram for this sequence of choices using A, B, and C to represent the children. There are six possible arrangements. b) Of the six arrangements that are possible, only one of them is the arrangement ABC. P(ABC) = 1/6

3 We dealt with arranging the names of three children to create sequences with different orders. In the simplest case, we used all the children's names to fill the same number of positions. This was done by making a first choice in which all names could be chosen. Then, the second choice had one fewer choice available, since one child's name and had already been selected and was no longer available. Finally, the last choice was forced since there was only one remaining name. Thus, there are 3 x 2 x 1 ways of putting the children in a line.

4 Factorial Notation Factorial notation presents us with a method of easily representing the expression included on the last slide

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7 Practice: a) 5! b) 4! 2! c) 8! x 2! d)

8 Factorials are useful when finding the number of ways of ordering all of the items available in a set. There will be times when we are looking to find the number of different ways of selecting and ordering a selection from the larger set. For example, lets say instead of having 3 children for 3 positions for a line-up, you now have to choose from 15 children to arrange into 3 positions. How many possibilities does he have?

9 Example: You have 15 children from which to choose and arrange 3 players. How many possibilities do you have?

10 Example: You and two of your good friends are in the final group of 12 candidates for three open jobs at Tim Hortons! They need one person to take orders, one person at the window, and one person to work in the kitchen. You have been told that management will randomly choose their three workers from the final group. What is the probability that you and your two friends will be hired? Solution: First, we need to determine the number of ways that 3 people can be chosen and ordered from the group of 12.

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13 Selecting Objects With and Without Replacement Sampling can be done with replacement or without replacement. When done with replacement, the selected object is put back before the next object is selected. When done with replacement, the event remain independent of each other, whereas if done without replacement, they become dependent.

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15 In groups you will present your answers to the class for the following questions: 1. Twelve students have signed up to serve on the yearbook committee. a) How many ways can the staff adviser choose three people to act as publisher, lead photographer, and editor, respectively? b) In filling these three positions, what is the probability that Frances is selected as publisher and Marc is chosen as editor? 2. The moderator of a form discussion is assigning seats to the six participants. If the seats are arranged in a linear fashion, determine the probability that Dr. Eisen and Dr. Bugada are seated next to each other. 3. How many three-letter arrangements are possible using the letters of the word CANADA? 4. Create a permutation question that also involves probability. Determine the solution to your problem.