chap2-3slidespt4

chap2-3slidespt4 - Permutations Definition Let A be a set...

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Permutations Deﬁnition Let A be a set containing n elements. Deﬁnition An r -element permutation of A is an ordered arrangement of r ( 0 < r n ) elements taken without replacement from A . The number of such permutations is denoted P ( n,r ) . ECSE 305 - Winter 2012 (slides based in part on the notes by B. Champagne) 106

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Permutations Example Consider the set A = { a,b,c } . All 2-element permutations are: start b a c b c a b c a b c a ab ac ba bc ca cb 6 = Total: P (3 , 2) = 6 . ECSE 305 - Winter 2012 (slides based in part on the notes by B. Champagne) 107
Permutations The number of r -element permutations of a set A containing n elements is given by the product P ( n,r ) = n ( n - 1) ... ( n - r + 1) Proof. x 1 x 2 ... x r n choices n - 1 choices ... n - r choices ECSE 305 - Winter 2012 (slides based in part on the notes by B. Champagne) 108

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Permutations Factorial notation Deﬁnition For any n N : n ! = n ( n - 1)( n - 2) ... 1 We also deﬁne 0! = 1 . Alternatively: n ! = n ( n - 1)! , with initial condition 0! = 1 . Finally: P ( n,r ) = n ! ( n - r )! ECSE 305 - Winter 2012 (slides based in part on the notes by B. Champagne) 109
Permutations Example How many diﬀerent words can we form with the 3 letters A, B, C? ( # of 3-letter words) = ( # of 3 -element permutations of { A, B, C } ) 3 × 2 × 1 = 3! = 6 ECSE 305 - Winter 2012 (slides based in part on the notes by B. Champagne) 110

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Permutations Example How many diﬀerent 2-letter words can we form with the 3 letters A, B, C? ( # of 2-letter words) = ( # of 2 -element permutations of { A, B, C } ) 3 × 2 = 6 ECSE 305 - Winter 2012 (slides based in part on the notes by B. Champagne) 111
Permutations Example How many diﬀerent words can we form with the 4 letters A, B, C, C? Complication: The two C’s are identical. What if we could distinguish amongst the two C’s? How many diﬀerent words can we form with the 4 letters A, B, C, C’? 4! = 24 ECSE 305 - Winter 2012 (slides based in part on the notes by B. Champagne) 112

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Permutations Example Of these 24 words some are equivalent: ACBC’ = AC’BC = ACBC . The are 2! pairs of words that are equivalent.
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chap2-3slidespt4 - Permutations Definition Let A be a set...

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