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Published byGwendoline Beasley Modified over 8 years ago
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Permutations
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Basics 5! Does not mean FIVE! 5! = 5 4 3 2 1 7! = 7 6 5 4 3 2 1 5! is read five factorial. 5! = 120 7! = 5040
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When a group of objects or things are arranged in a certain order, the arrangement is called a permutation. The arrangement of objects in a line is called a linear permutation.
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Example 1. A theater owner has 11 films to show on 8 different screens. How many different arrange- ments are there for the 8 screens to show the 11 movies.
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Example 1. 8 screens and 11 movies. We must make 8 choices. 1234 5678 111098 7654 Choice Choice # #
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Example 1. 1234 5678 111098 7654 Choice Choice # # Therefore by the FCP there are 11 10 9 8 7 6 5 4
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Example 1. Therefore by the FCP there are 11 10 9 8 7 6 5 4 arrangements. The number of way to arrange 11 objects taken 8 at a time is written as P(11,8).
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P(n,r) is read as the permutation of n objects taken r at a time. P(n,r) = Definition of P(n,r) n! n!(n-r)!
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Example 2. P(7,5) = Five teens find seven empty seats at a theater. How many different seating arrangements are there? 7! 7!(7-5)! 7!2!=
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Example 2. P(7,5) = 7! 7!(7-5)! 7!2!= 7 6 5 4 3 2 1 2 1 2 1= = 7 6 5 4 3 = 2520
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Example 3. A music store manager want to arrange 5 rock CD’s 4 rap CD’s and 4 jazz CD’s on a shelf. How many ways can they be arranged if they are ordered according to type?
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Example 3. The 5 rock CD’s arrange into P(5,5) or 5! ways. The 4 rap CD’s arrange into P(4,4) or 4! ways. The 4 jazz CD’s arrange into P(4,4) or 4! ways.
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Example 3. There are 3 types of CD’s that arrange into P(3,3) or 3! ways. The total ways the CD’s can be arranged is the product. P(5,5)P(4,4)P(4,4)P(3,3) = 5!4!4!3! = 414,720
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Permutations with repetitions. The number of permutations of n objects of which p are alike and q are alike is n! n! p!q! p!q! The rule can be extended to any number of objects that repeat.
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Example 4. How many ways can the letters of the word perpendicular be arranged? There are 2 p’s, 2 e’s, and 2 r’s. Therefore 13! 13!2!2!2!arrangements
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Example 4. How many ways can the letters of the word perpendicular be arranged? 13! 13!2!2!2! = 778,377,600
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Circular Permutations. If n distinct objects are arranged in a circle there are or (n-1)! permutations of the objects around the circle. n!n
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Example 5. A disc jockey is loading a circular tray with six compact discs. How many different ways can these be arranged? (n-1)! = (6-1)! = 5! = 120
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Circular Permutations (cont). If n distinct objects are arranged in a circle and there is a fixed point on the circle then there are n! permutations of the objects around the circle.
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Example 6. If five different homes are being built around a cul-de-sac. How many different arrangements of the homes are possible? The cul-de-sac is circular but the entrance road is a fixed point
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Example 6. If five different homes are being built around a cul-de-sac. The cul-de-sac is circular but the entrance road is a fixed point There are n! = 5! =120 arrangements possible.
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We now have several formulas to remember. Linear permutations, circular permutations, circular permutations with a fixed point, and permutations with repetitions.
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One last thing we need to know. Reflection happens when a circular permutation can be flipped over or when a linear permutation is viewed from opposite sides. This causes the number of permutations to be half as many.
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