Unformatted text preview: ± 1 2 ² : For example, ± 5 2 ² D 4 C ± 4 2 ² D 4 C 3 C ± 3 2 ² D 4 C 3 C 2 C ± 2 2 ² D 4 C 3 C 2 C 1 D 10: In general, ± n 2 ² D .n ± 1/ C .n ± 2/ C ²²² C 2 C 1: This sum is well known and equal to n.n ± 1/=2 . (You can ﬁnd an inductive proof of the formula .n ± 1/ C .n ± 2/ C ²²² C 2 C 1 D n.n ± 1/=2 in Appendix C.1 .) Here is another argument for the formula ± n 2 ² D n.n ± 1/=2 that is better because it generalizes. Think of building the nŠ permutations of f 1;2;:::;n g in the following way. First choose two elements to be the ﬁrst two (leaving n ± 2 to be the last n ± 2 ). This can be done in ± n 2 ² ways....
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 Fall '08
 EVERAGE
 Algebra, Sets, Counting, Natural number, 5g, å å, two–element subsets

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