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Unformatted text preview: commutative rings.) 1.30 Show that the binomial coef f cients are symmetric : n r = n n r for all r with 0 r n . Solution. By Lemma 1.17, both ( n r ) and ( n n r ) are equal to n ! r ! ( n r ) ! . 1.31 Show, for every n , that the sum of the binomial coef f cients is 2 n : n + n 1 + n 2 + + n n = 2 n . Solution. By Corollary 1.19, if f ( x ) = ( 1 + x ) n , then there is the expansion f ( x ) = n + n 1 x + n 2 x 2 + + n n x n . Evaluating at x = 1 gives the answer, for f ( 1 ) = ( 1 + 1 ) n = 2 n ....
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 Fall '11
 KeithCornell

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