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Counting, Permutations, & Combinations A counting problem asks “how many ways” some event can occur. Ex. 1: How many three-letter codes are there using letters A, B, C, and D if no letter can be repeated? • One way to solve is to list all possibilities. Ex. 2: An experimental psychologist uses a sequence of two food rewards in an experiment regarding animal behavior. The food rewards consists of three options. How many different sequences of rewards are there if each option can be used only once in each sequence? Next slide •Another way to solve is a factor tree where the number of end branches is your answer. b a c a b c a c b Fundamental Counting Principle • If there are m ways for Event A to occur and n ways for Event B to occur, then there are m × n ways for Events A and B to occur together. Fundamental Counting Principle Suppose you’re trying to figure out the number of possible license plate numbers in the state. Numbers and letters are possible. Create 7 “slots” -> _ _ _ _ _ _ _ Ask if repeats are possible. Yes, they are. Ask if order matters. Yes, it does. Therefore 36 36 36 36 36 36 36 = 367 = 7.836 x 1010 = 78,364,164,100 Ex. 2: What if each number and letter was only allowed to be used once? Using the fundamental counting principle but saying that repeats are NOT allowed AND order still matters: 36 35 34 33 32 31 30 = 4.207 x 1010 This set of criteria (no repeats but order matters) is also called a permutation Permutations An r-permutation of a!set of nfactorial means elements is an ordered Ex. selection of r 3! = 3∙2∙1 elements from the set of n elements 0! = 1 n! n Pr n r ! Ex. 1:How many three-letter codes are there using letters A, B, C, and D if no letter can be repeated? Note: The order does matter 4! 24 4 P3 1! Combinations The number of combinations of n elements taken r at a time is n! n Cr r!n r ! Order does matter! WhereNOT n & r are nonnegative integers & r < n Ex. 3: How many committees of three can be selected from four people? Use A, B, C, and D to represent the people Note: Does the order matter? 4! 4 4 C3 3!1! Ex. 4: How many ways can the 4 call letters of a radio station be arranged if the first letter must be W or K and no letters repeat? 2 25 24 23 27,600 Ex. 5: In how many ways can our class elect a president, vicepresident, and secretary if no student can hold more than one office? n P3 Ex. 6: How many five-card hands are possible from a standard deck of cards? 52 C5 2,598,960 Ex. 7: Given the digits 5, 3, 6, 7, 8, and 9, how many 3-digit numbers can be made if the first digit must be a prime number? (can digits be repeated?) Think of these numbers as if they were on tiles, like Scrabble. After you use a tile, you can’t use it again. 3 5 4 60 Ex. 8: In how many ways can 9 horses place 1st, 2nd, or 3rd in a race? 9 P3 504 Ex. 9: In how many ways can a team of 9 players be selected if there are 3 pitchers, 2 catchers, 6 in-fielders, and 4 out-fielders? 3 C1 2 C1 6 C4 4 C3 360