lec17-permutations - m and the other gives n ± n r ² = ±...

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EECS 203, Winter 2007 Discrete Mathematics Lecture 17 Permutations and combinations March 8 Reading: Rosen [5.3] March 8 Permutations and combinations, Page 1
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17.1 Permutations An r -permutation of a set A is an ordered arrangement of r elements from A . If | A | = n , the number of r -permutations of A is P ( n, r ) = n ( r ) = n ( n - 1) · · · ( n - r + 1) . Examples A ballroom dancing party. Bus. Chess. March 8 Permutations and combinations, Page 2
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17.2 Combinations A r -combination of a set A is a subset of A of r elements. If | A | = n , the number of r -combinations is C ( n, r ) = n r = P ( n, r ) /r ! = n ! r !( n - r )! . Examples Football team. How many outcome sequences of tossing n coins contain precisely m heads? Contains precisely m heads and they occur consecutively? March 8 Permutations and combinations, Page 3
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17.3 Combinatorial proof Combinatorial Proof: a proof for an identity m = n by counting in two ways the same set of objects so that one way gives
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Unformatted text preview: m and the other gives n . ± n r ² = ± n n-r ² . ± 2 n n ² = n X r =0 ± n r ² 2 . March 8 Permutations and combinations, Page 4 17.4 Division rule If f : A → B is a k-to-1 function, and A and B are finite sets, then | A | = k | B | . Another chess problem How many arrangements of the letters BOOK? March 8 Permutations and combinations, Page 5 17.5 Poker hands A hand is a 5-combination. How many hands are: • Four-of-a-Kind: include a set of cards of the same value. • Full-House: three cards of one value and two cards of another value. • Two-Pairs: two cards of one value, two of another value, and one of a third value. • Every-Suit: contains at least one card from every suit. March 8 Permutations and combinations, Page 6...
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