Discrete Mathematics with Graph Theory (3rd Edition) 209

Discrete - f that is assume l(f £-1(f = xHl L k xHl-kyk L k xl-kyk l yHl k=1 k=O = xHl t G)xHl-kyk t(k ~ 1)xHl-kyk yHl = xHl t G(k ~ 1 xHl-kyk yHl

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Section 7.7 207 hypothesis). This sum is (ktl), by (5). The third number is the sum of the numbers to the left and right immediately above it; that is, (~) + m. This sum is (ktl) by (5). In general, for any r, 1 < r :::; k + 1, the rth number is the sum of the numbers to the left and right immediately above it. Using the induction hypothesis and the identity in 5, this sum is (r~2) + (r~l) = (;:i). So the numbers in row k + 1 are the binomial coefficients, as asserted. 18. If n = 1, x + y = @xlyO + G)xOyl, as required. Assume the result is true for n =
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Unformatted text preview: f; that is, assume l (f) £-1 (f) = xHl + L k xHl-kyk + L k xl-kyk+l + yHl k=1 k=O = xHl + t. G)xHl-kyk + t. (k ~ 1)xHl-kyk + yHl = xHl + t. [ G) + (k ~ 1) ] xHl-kyk + yHl by identity (5) HI = '"" (f + 1) Hl-k k ~ k x y. k=O By the Principle of Mathematical Induction, the proof is complete. 19. (a) t, G) = G) + G) + G) + G) + (~) = 1 + 3 + 6 + 10 + 15 = 35 = G)' The sum here is the sum of the entries of Pascal's triangle on the third southwest-northeast diago-nal (f) beginning with the entry above and to the left of 35 = G) and moving up and right....
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This note was uploaded on 11/08/2010 for the course MATH discrete m taught by Professor Any during the Summer '10 term at FSU.

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