InterviewSolution
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InterviewSolution
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Statement-1 `sum_(r=0)^(n) r ""^(n)C_(r) x^(r) (-1)^(r) = nx (1 - x)^(n -1)` Statement-2:` sum_(r=0)^(n)r ""^(n)C_(r) x^(r) (-1)^(r) =0`A. 1B. 2C. 3D. 4 |
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Answer» Correct Answer - d We have, `sum_(r=0)^(n) r ""^(n)C_(r) x^(r) (-1)^(r) = sum_(r=0)^(n) (-r)(n)/(r) ""^(n-1)C_(r-1) x^(r-1)x(-1)^(r-1)` `rArr sum_(r=0)^(n) r ""^(n)C_(r) x^(r) (-1)^(r) = -nx sum_(r=1)^(n) ""^(n-1)C_(r-1) x^(r-1)x(-1)^(r-1)` `rArr sum_(r=0)^(n) r ""^(n)C_(r) x^(r) (-1)^(r) = -nx (1 -x)^(n-1)` ...(i) So, statement-1 is not true. Replacing x by 1 in (i) , we get ` sum_(r=0)^(n) r ""^(n)C_(r)(-1)^(r) =0` So, statement-2 is true. |
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