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Lecture on Permutation Combination

# Lecture on Permutation Combination - Section 6.3 1...

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Section 6.3 1

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2 Section Summary y Permutations y Combinations y Combinatorial Proofs
3 Permutations Definition : A permutation of a set of distinct objects is an ordered arrangement of these objects. An ordered arrangement of r elements of a set is called an r permuation . Example : Let S = { , , }. y The ordered arrangement 3 , 1 , 2 is a permutation of S . y The ordered arrangement 3 , 2 is a 2 permutation of S . y The number of r permuatations of a set with n elements is denoted by P ( n , r ). y The 2 permutations of S = { 1 , 2 , 3 } are 1 , 2; 1 , 3; 2 , 1; 2 , 3; 3 , 1; and 3 , 2. Hence, P ሺ3,2ሻ ൌ 6.

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4 A Formula for the Number of Permutations Theorem 1 : If n is a positive integer and r is an integer with 1൑ r n , then there are P ( n , r ) = n ( n െ1 )( n െ2 ) ··· ( n r + 1 ) r permutations of a set with n distinct elements. Proof : Use the product rule. The first element can be chosen in n ways. The second in n െ 1 ways, and so on until there are n ( r )) ways to choose the last element. y Note that P ( n , 0 ) = 1 , since there is only one way to order zero elements. Corollary 1 : If n and r are integers with r n, then
5 Solving Counting Problems by Counting Permutations Example : How many ways are there to select a first prize winner, a second prize winner, and a third prize winner from different people who have entered a contest? Solution : P( , ) = =

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6 Solving Counting Problems by Counting Permutations Example : Suppose that a saleswoman has to visit eight different cities. She must begin her trip in a specified city, but she can visit the other seven cities in any order she wishes. How many possible orders can the saleswoman use when visiting these cities? Solution : The first city is chosen, and the rest are ordered arbitrarily. Hence the orders are: = =
7 Solving Counting Problems by Counting Permutations Example : How many permutations of the letters ABCDEFGH contain the string ABC ? Solution : We solve this problem by counting the permutations of six objects, ABC , D , E , F , G , and H . = =

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8 Combinations Definition : An r combination of elements of a set is an unordered selection of r elements from the set. Thus, an r combination is simply a subset of the set with r elements. y The number of r combinations of a set with n distinct elements is denoted by C ( n , r ). The notation is also used and is called a binomial coefficient . Example : Let S be the set { a , b , c , d }. Then { a , c , d } is a 3 combination from S. It is the same as { d , c , a } since the order listed does not matter. y C ( 4 , 2 ) = 6 because the 2‐combinations of { a , b , c , d } are the six subsets { a , b }, { a , c }, { a , d }, { b , c }, { b , d }, and { c , d }.
9 Combinations Theorem : The number of r combinations of a set with n elements, where n r P ( n , r ) = C ( n , r ) P ( r , r ). Therefore,

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10 Combinations Example : How many poker hands of five cards can be dealt from a standard deck of 52 cards? Also, how many ways are there to select 47 cards from a deck of 52 cards?
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Lecture on Permutation Combination - Section 6.3 1...

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