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Tree diagram for the experiment of tossing two coins start H T H H T T

Counting Techniques Tree Diagram Multiplication Rule Permutations Combinations Tree Diagram a method of listing outcomes of an experiment consisting of a series of activities Tree diagram for the experiment of tossing two coins start H T H H T T Find the number of paths without constructing the tree diagram: Experiment of rolling two dice, one after the other and observing any of the six possible outcomes each time. Number of paths = 6 x 6 = 36 Multiplication of Choices If there are n possible outcomes for event E 1 and m possible outcomes for event E 2, then there are n x m or nm possible outcomes for the series of events E 1 followed by E 2. Area Code Example Until a few years ago a three-digit area code was designed as follows. The first could be any digit from 2 through 9. The second digit could be only a 0 or 1. The last could be any digit. How many different such area codes were possible? 8 2 10 = 160 Ordered Arrangements In how many different ways could four items be arranged in order from first to last? 4 3 2 1 = 24 Factorial Notation n! is read "n factorial" n! is applied only when n is a whole number. n! is a product of n with each positive counting number less than n Calculating Factorials 5! = = 3! = = 120 6 Definitions 1! = 1 0! = 1 Complete the Factorials: 4! = 10! = 6! = 15! = 24 3,628, x 10 12 Permutations A permutation is an arrangement in a particular order of a group of items. There are to be no repetitions of items within a permutation.) Listing Permutations How many different permutations of the letters a, b, c are possible? Solution: There are six different permutations: abc, acb, bac, bca, cab, cba. Listing Permutations How many different two-letter permutations of the letters a, b, c, d are possible? Solution: There are twelve different permutations: ab, ac, ad, ba, ca, da, bc, bd, cb, db, cd, dc. Permutation Formula The number of ways to arrange in order n distinct objects, taking them r at a time, is: Another notation for permutations: Find P 7, 3 Applying the Permutation Formula P 3, 3 = _______ P 4, 2 = _______ P 6, 2 = __________ P 8, 3 = _______ P 15, 2 = _______ Application of Permutations A teacher has chosen eight possible questions for an upcoming quiz. In how many different ways can five of these questions be chosen and arranged in order from #1 to #5? Solution: P 8,5 = = = 6720 Combinations A combination is a grouping in no particular order of items. Combination Formula The number of combinations of n objects taken r at a time is: Other notations for combinations: Find C 9, 3 Applying the Combination Formula C 5, 3 = ______ C 7, 3 = ________ C 3, 3 = ______ C 15, 2 = ________ C 6, 2 = ______ Application of Combinations A teacher has chosen eight possible questions for an upcoming quiz. In how many different ways can five of these questions be chosen if order makes no difference? Solution: C 8,5 = = 56