AP Stats BW 10/14 or 15 Suppose that the numbers of unnecessary procedures recommended by five doctors in a 1-month period are given by the set {2, 2, 8, 20, 33}. If it is discovered that the fifth doctor recommended an additional 25 unnecessary procedures, how will the median and mean be affected?

Section 3.4 – Counting Principles SWBAT: Use Permutations and Combinations in finding probabilities. Source:

Counting Principles In Section 3.1, we learned the Fundamental Counting Principle is used to find the number of ways two or more events can occur in sequence. In this section we will study several other techniques for counting the number of ways an event can occur.

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Permutations: The number of permutations of n distinct objects taken r at a time is n P r =, where r < n EX:Four people went to the movies and there are four open seats next to each other in the middle of the movie theatre. How many ways can they arrange themselves? Using Fundamental Counting Principle: ___ _ * _____ * _____ * ____ Seat1Seat2Seat3Seat4 4 * 3 * 2 * 1 = 24 Using Permutations: n P r = = 4 P 4 = = 24 ways

Permutations, Cont’d EX:How many arrangements are there of the word ANGLE? Using Fundamental Counting Principle: 5! = 5 * 4 * 3 * 2 *1 = 120 Using Permutations: 5 P 5 = EX:What if four people go to a movie, but there are only two open seats next to each other? How many distinct ways can these four people arrange themselves in the two seats? Using Fundamental Counting Principle: 4 * 3 = 12 4 P 2 = Using Permutations:

You Try…. Use Permutations to find… EX:Six people are running for four different offices on the school board. In how many different ways can those offices be filled? EX:Twelve horses are running in a race. How many different ways can 1 st, 2 nd, and 3 rd place be awarded? 6 P 4 = 360 ways 12 P 3 = 1,320 ways EX:How many 3-letter words can be formed from the “word” TEXAS if each letter is used only once in a “word”? 5 P 3 = 60 ways

Let’s try a trickier one… a)In how many ways can the letters A, B, C, D and E be arranged in a row? b)In how many of these arrangements is D always the fist? 5 P 5 = 120 ways 1 * 4 * 3 * 2 * 1 = 24 4 P 4 = 24 ways

Distinguishable Permutations: Order a group of n objects in which some objects are the same EX:How many ways can I arrange the letters AAAABBC?

You Try…. EX:How many different ways can I arrange the letters in the words below? a)APPLE b)DEGREE c)DIVIDED

You Try…. EX: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: A selection of r objects from a group of n objects without regard to order n C r = EX:I select two people from class to go to Hawaii. I select 3 toppings on a pizza. Vs. Choosing a 2 digit password…. Order would matter, so it would be a permutation

Combinations, cont’d EX:You have five choices of sandwich fillings. How many different sandwiches could you make by choosing three of the five fillings? 5 C 3 = EX:Katie is going to adopt kitten from a litter of eleven. How many ways can she choose a group of 3 kittens? 11 C 3 =

Let’s try a trickier one… There are fourteen juniors and twenty-three seniors in the Service Club. The club is to send four representatives to the State Conference. a)How many ways are there to select a group of four students to attend the conference? b)If members of the club decide to sent two juniors and two seniors, how many different groupings are possible? 37 C 4 = 66,045 ways 14 C 2 * 23 C 2 = 23,023 ways

You try….. Tell whether each situation represents a permutation or a combination? a.A coach chooses a team of 6 players from 12. b.Ten people are in a line to buy tickets. c.A teacher selects a committee of 4 students from 25 students. d.The different groups of three vegetables you could choose from six different vegetables. e.Different orders you can play 4 DVDs. f.Different groups of class officers students can elect from a class of 25 students. a.C b.P c.C d.C e.P f.P

One more… A student advisory board consists of 17 members. Three members serve as the board’s chair, secretary, and webmaster. Each member is equally likely to serve any of the positions. What is the probability of selecting at random the three members that hold each position. a)Find the number of ways the three positions can be filled. b)Find the probability of correctly selecting the three members 17 P 3 = 4080 ways

OK…really…One more… Sue Bartling loves to read mystery books and car-repair manuals. On a visit to the library, Sue finds 9 new mystery books and 3 car-repair manuals. She borrows 4 of these books. Find the number of different sets of 4 books Sue can borrow if: a)All are mystery books b)Exactly 2 are mystery books c)Only one is a mystery book 9 C 4 = 126 different sets 9 C 2 * 3 C 2 = 108 different sets 9 C 1 * 3 C 3 = 9 ways NOTE: These have to add up to 4

You try….. Find the probability of being dealt five diamonds from a standard deck of playing cards. -All 5 together, order does not matter (group of 5) -# of ways of choosing 5 diamonds: 13 C 5 -# of possible 5 card hands: 52 C 5 - Probability: 13 C 5 / 52 C 5 =

HOMEWORK: Worksheets: 3.4 Practice: Combinations & Permutations 3.4 – Applications of the Counting Principle