3.7: Counting Objective: To find the counts of various combinations and permutations, as well as their corresponding probabilities CHS Statistics

Warm-Up Alfred is trying to find an outfit to wear to take Beatrice on their first date to Burger King. How many different ways can he make an outfit out of this following clothes: Pants: Green, Baby Blue, Black, Grey Shirt: Red, Pink, Plaid, Blue, Lime Green Tie: Polka dot, Stripped

Fundamental Counting Principle For a sequence of two events in which the first event can occur in m ways and the second event can occur in n ways, the events together can occur a total of m n ways. Example: You are purchasing a new car. The possible manufacturers, car sizes, and colors are listed. Manufacturer: Ford, GM, Honda Car Size: compact, midsize Color: White, red, black, and green How many different ways can you select one manufacturer, one car size, and one color?

Fundamental Counting Principle (cont.) Example: The access code for a garage door consists of four digits. How many codes are possible if: Each digit can be used only once and not repeated? Each digit can be repeated? Each digit can be repeated but the first digit cannot be 8 or 9?

Factorial Rule Examples: How many ways can 5 people be seated on a bench? How many ways can a class of 50 be ranked by grades? To answer questions like these, we will use the factorial rule. Factorial Rule A collection of n different items can be arranged in order n! different ways. n! = n x (n – 1) x (n – 2) x (n – 3) x … 5! = 9! = 2! =

Factorial Rule (cont.) Examples: How many ways can 5 people be seated on a bench? How many ways can a class of 50 be ranked by grades?

Permutations Example: Forty-three sprinters race in a 5K. How many ways can they finish first, second, and third? Can we use the factorial rule? Why or why not?

Permutations (When all Items Are Different) Permutations: When r items are selected from n available items (without replacement). Therefore, the order matters. Calculate the following permutations:

Permutations (cont.) Example: Forty-three sprinters race in a 5K. How many ways can they finish first, second, and third?

Distinguishable Permutations

Distinguishable Permutations (cont.) Example: A building contractor is planning to develop a subdivision. The subdivision is to consist of 6 one-story houses, 4 two-story houses, and 2 split-level houses. In how many distinguishable ways can the houses be arranged?

Combinations Example: You are picking 3 different flavors to put on your banana split. You can choose from 25 different flavors. How many ways can this be done? Does the order matter here?

Combinations Combination Rule: When order does not matter, and we want to calculate the number of ways (combinations) r items can be selected from n different items. RECAP: When different orderings of the same items are counted separately, we have a permutation problem, but when different orderings of the same items are not counted separately, we have a combination problem.

Combinations (cont.) Calculate the following combinations: Example: You are picking 3 different flavors to put on your banana split. You can choose from 25 different flavors. How many ways can this be done? Example: You want to buy three different CDs from a selection of 5 CDs. How many ways can you make your selection?

Combinations (cont.) Example: A states department of transportation plans to develop a new section of interstate highway and receives 16 bids. The state plans to hire four of the companies. How many different ways can the companies be selected? Example: The manager of an accounting department want to form a three-person advisory committee from the 20 employees in the department. In how many ways can the manager form this committee?

Probability Using Permutation and Combination A student advisory board consists of 17 members. Three members serve as the boards chair, secretary, and webmaster. What is the probability of selecting at random the three members that will hold these positions? You have 11 letters consisting of one M, four Is, four Ss, and two Ps. If the letters are randomly arranged in order, what is the probability that the arrangement spells the word Mississippi?

Probability Using Permutation and Combination (cont.) Find the probability of being dealt five diamonds from a standard deck of playing cards? A food manufacturer is analyzing a sample of 400 corn kernels for the presence of a toxin. In this sample, three kernels have dangerously high levels of the toxin. If four kernels are randomly selected from the sample, what is the probability that exactly one kernel contains a dangerously high level of the toxin?

Probability Using Permutation and Combination (cont.) A jury consists of five men and seven women. Three are selected at random for an interview. Find the probability that all three are men?

Assignment 3.7 Practice