InterviewSolution
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InterviewSolution
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Stetemet - 1: `sum_(r=0)^(n) r. ""^(n)C_(r) = n 2^(n-1)` Statement-2: ` sum_(r=0)^(n) r. ""^(n)C_(r) x^(r) = n (1 + x )^(n-1) x`A. 1B. 2C. 3D. 4 |
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Answer» Correct Answer - a We have, `sum_(r=0)^(n) r. ""^(n)C_(r) x^(r) = sum_(r=0)^(n) r (n)/(r) ""^(n-1)C_(r-1) x^(r-1) xxx` `rArr sum_(r =0)^(n) r ""^(n)C_(r) x^(r) = nx sum_(r=1)^(n) ""^(n-1)C_(r-1) x^(r -1)` `rArr sum_(r=0)^(n) r ""^(n)C_(r) x^(r) = nx (1 + x)^(n-1)` ...(i) So, statement-2 is true. Replacing x by in (i), we get `sum_(r=0)^(n) r ""^(n)C_(r) = n2^(n-1)` So, statement -1 is also true stetement -2 is a correct expansion for statement-1. |
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