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Permutations Objectives: Solve problems involving linear permutations of distinct or indistinguishable objects. Solve problems involving circular permutations. Standards: 2.7.8A Determine the number of permutations for an event. A permutation is an arrangement of objects in a specific order. When objects are arranged in row, the permutation is called a linear permutation. You can use factorial notation to abbreviate this product: 4! = 4 x 3 x 2 x 1 = 24. If n is a positive integer, then n factorial, written n!, is defined as follows: n! = n x (n-1) x (n-2) x . . . x 2 x 1. Note that the value of 0! = 1. I. Permutations of n Objects - the number of permutations of n objects is given by n! {factorial button – go to Math to PRB to # 4} Ex 1. In 12-tone music, each of the 12 notes in an octave must be used exactly once before any are repeated. A set of 12 tones is called a tone row. How many different tone rows are possible? 12! = 479,001, 600 Ex 2. How many different ways can the letters in the word objects be arranged? 7! = 5040 II. Permutations of n Objects Taken r at a Time – the number of permutations of n objects taken r at a time, denoted by P(n, r), is given by P(n, r) = nPr =__n!_, where r < n. (n–r)! Ex 1. Find the number of ways to listen to 5 different CDs from a selection of 15 CDs. 15 P 5 = 360, 360 Ex 2. Find the number of ways to listen to 4 CDs from a selection of 8 CDs. 8 P 4 = 1680 Ex 3. Find the number of ways to listen to 3 different CDs from a selection of 5 CDs. 5 P 3 = 60 III. Permutations with Identical Objects – the number of distinct permutations of n objects with r identical objects is given by n!/r! where 1 < r < n. The number of distinct permutations of n objects with r1 identical objects, r2 identical objects of another kind, r3 identical objects of another kind, . . . , and rk identical objects of another kind is given by _______n! _. r1 ! * r2 ! * r3 ! . . . rk ! Ex 1. Anna is planting 11 colored flowers in a line. In how many ways can she plant 4 red flowers, 5 yellow flowers, and 2 purple flowers? Ex 2. In how many ways can Anna plant 11 colored flowers if 5 are white and the remaining ones are red? 11!__ (5! * 6!) = 462 Ex 3. Frank is organizing sports equipment for the physical education room. He has 15 balls that he must place in a line. In how many ways can he line up 6 footballs, 2 soccer balls, 4 kickballs, and 3 basketballs? ____15!______ (6! * 2! * 4! * 3!) = 6,306,300 Ex. 4 BETWEEN 7! 3! = 840 III. Circular Permutations - If n distinct objects are arranged around a circle, then there are (n – 1)! Circular permutations of the n objects. Ex 2. In how many ways can seats be chosen for 12 couples on a Ferris wheel that has 12 double seats? (12 – 1)! = 11! = 39, 916, 800 Ex 3. In how many different ways can 17 students attending a seminar be arranged in a circular seating pattern? (17 – 1)! = 16! = 2.09 X 1013 Writing Activities REVIEW OF PERMUTATIONS