Coaltions and Caucuses

Coaltions and Caucuses - COALITIONS AND CAUCUSES PAGE 1...

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C OALITIONS AND C AUCUSES P AGE 1 5/7/2009 Mathematical Background A set of n objects contains exactly 2 n subsets. For example, the 8 2 3 = subsets of { } c b, a, are { } { } { } { } { } { } { } { } . c b, a, , c b, , c a, , b a, , c , b , a , We have arranged the eight subsets in order of the number of elements they contain: first the one subset of size zero, then the three subsets of size one, then the three subsets of size two, and finally the one subset of size three. Notice that half of the subsets listed above contain at most two letters, and half contain at least three letters. In general, if n is an odd number, then exactly half of the 2 n subsets of a set of n objects contain less than 2 n objects, and half contain more than 2 n objects. So, for example, 10 2 of the 11 2 subsets of { } k j, i, h, g, f, e, d, c, b, a, contain at most five letters, and the other 10 2 of them contain at least six letters. In general, a set of size n contains exactly ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ! ! ! 1 2 1 1 2 1 factors factors k k n n k k k k n n n n k n k k - = - - ÷ + - - - = subsets of size k , where the symbol n ! stands for the product of all the positive integers from 1 to n . For example, the number of subsets of { } c b, a, of size two was seen to be ( 29 ( 29 ( 29 . 3 2 1 6 ! 2 ! 1 ! 3 1 2 2 3 2 3 = = = ÷ = Similarly, ( 29 ( 29 . 120 ! 7 ! 3 ! 10 1 2 3 4
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Coaltions and Caucuses - COALITIONS AND CAUCUSES PAGE 1...

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