11 1 Permutations and Combinations You just bought three pairs of pants and two shirts. How many different outfits can you make with these items? Using a tree diagram, you can see that you can make six different outfits. It is not always feasible to use a tree diagram, especially when there are lots of choices to be made. Instead, you can use the Fundamental Counting Principle. Fundamental Counting Principle If an event M can occur in m ways and is followed by an event N that can occur in n ways, then the event M followed by the event N can occur in m times n ways. Simply multiply the ways the events can occur. 1
1. A deli has a lunch special which consists of a sandwich, soup, dessert and drink for $4.99. They offer the following choices: Sandwich chicken salad, ham, tuna, roast beef Soup tomato, vegetable, chicken noodle Dessert cookie, pie Drink tea, coffee, Coke, Diet Coke, Sprite How many possible lunch specials are there? 2. You are taking a True/False test with five questions. If you answer each question with T or F and leave none blank, in how many ways can you answer the whole test? 2
3. A company places a six symbol code on each unit of product. The code consists of four digits, the first of which is the number five, followed by two letters, the first of which is not a vowel. How many different codes are possible? 4. An employee of a company has a six digit pin number to access the work site. The code may not begin with a zero, and no digit may repeat. How many different codes are possible? 3
5. In how many orders can eight actors be listed in the opening credits of a movie if the leading actor must be listed first or last? permutation: an arrangement of items in a particular order You can use the Fundamental Counting Principle or the following formula. or where n is the number of distinct objects placed r at a time 6. In how many ways can eight CD s be arranged on a shelf? 7. If a softball league has ten teams, how many different end of season rankings are possible? (Assume no ties.) 8. In how many ways can a sorority of twenty members select a President, Vice President and Treasurer, assuming no one can hold more than one office? May 14 11:48 AM 4
combination: a selection in which order does not matter or where n is the number of distinct objects placed r at a time 9. You are going to draw four cards from a deck of 52 cards. How many four card hands are possible? 10. You have twenty songs on your MP3 player. You have time to listen to four of the songs. In how many different ways can you select four songs if order does not matter? Apr 16 10:35 AM (p. 681) 11 2 Probability (p. 681) 1. Apr 16 11:14 AM 5
sample space: the set of all possible outcomes to an experiment or activity equally likely outcomes: when each outcome in a sample space has the same chance of occurring (like tossing a coin or rolling a die) (p. 683) You can think of it like this... Theoretical Probability of an Event = Number of favorable outcomes Number of total outcomes Apr 16 11:18 AM Example A bag of M&M s contains 8 brown, 6 green, 5 red, 5 orange and 6 yellow candies. Find each probability or odds. 2. If one candy is chosen at random find each. a. P(1 green) b. P(red) c. P(not brown) Apr 16 11:23 AM 6
When working with probability, first decide if the problem is a simple probability or a combination. Simple Probability chooses one item at a time Combination chooses multiple items at a time Example A bag of M&M s contains 8 brown, 6 green, 5 red, 5 orange and 6 yellow candies. Find each probability or odds. 3. If 3 candies are selected at once randomly, find each. a. P(3 red) b. P(3 brown) combinatorics: includes using the Fundamental Counting Principle and ways to count permutations and combinations to find theoretical probability c. P(2 red, 1 yellow) 4. If 9 candies are selected at once randomly, find P(1red, 2 orange, 6 green) Apr 16 11:16 AM 5. (p. 683) Apr 16 11:31 AM 7
6. (p. 684) Apr 16 11:33 AM 7. (p. 684) Apr 16 11:33 AM 8
Practice (p. 678) 5, 9, 10, 11, 20, 29, 38 41, 48 50 (p. 685) 1, 2, 8, 9, 17 33 Apr 16 11:34 AM 9