Unformatted text preview: commutative rings.) 1.30 Show that the binomial coef f cients are “ symmetric ” : ³ n r ´ = ³ 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 : ³ n ´ + ³ n 1 ´ + ³ n 2 ´ + ··· + ³ n n ´ = 2 n . Solution. By Corollary 1.19, if f ( x ) = ( 1 + x ) n , then there is the expansion f ( x ) = ³ n ´ + ³ n 1 ´ x + ³ n 2 ´ x 2 + ··· + ³ 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
 Binomial Theorem, Addition, Complex Numbers

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