307_combinatoric_rules

307_combinatoric_rules - Summary of our combinatorial rules...

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Unformatted text preview: Summary of our combinatorial rules Here are our main rules for counting sets in connection with finite sample spaces with equally likely outcomes. Multiplicative Rule: Suppose each element of a set can be viewed as a choice, and we can organize this choice into two stages. Suppose that the first stage choice can be made In m ways, and for each first stage choice the second stage choice can be made in n ways. Then the overall choice can be made in mn ways, and hence the set has mn elements. This rule of course can be extended to the case where the overall choice can be organized into k stages. Strings: suppose a set consists of n symbols. Then the number of strings of k symbols from these n symbols (repetition ALLOWED) is n k . Permutations: Suppose a set consists of n symbols. Then a permutation of k elements from these n is a string of k of the n symbols, repetition NOT ALLOWED. The number of permutations of k elements from n is nPk = n ( n − 1)… ( n − k + 1) = n! . ( n − k )! Combinations: Suppose a set consists of n elements. A combination of k of the n elements is simply a subset of k of the n elements. (So order of selection is irrelevant.) The number of combinations of k of the n elements is given by the binomial coefficient nCk = ⎜ ⎟ = ⎛n⎞ ⎝k ⎠ n! . . k !( n − k ) ! Note: an alternative formulation of this situation is that nCk is the number of distinguishable ways to classify n elements into two categories, k elements in the first category and n‐k elements in the second. A classification amounts to specifying which particular elements are in which category. Multinomial coefficients: The number of distinguishable ways to classify n elements into k categories, with n1 elements into the first category, n2 elements in the second category, etc, where n1 + n2 + … + nk = n, is given by the multinomial coefficient n! . n1 !n2 !… nk ! Balls into boxes: The number of ways to put n indistinguishable balls into k boxes is given by ⎛ n + k − 1⎞ ⎜ ⎟ . Note: since such a “way” is determined by how many balls go into each box, this is also ⎝ k −1 ⎠ the number of solutions to n1 + n2 + … + nk = n in nonnegative integers n1 , n2 ,… , nk . ...
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This note was uploaded on 06/12/2010 for the course MATH 307 taught by Professor Luikonnen during the Fall '08 term at Tulane.

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