InterviewSolution
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InterviewSolution
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Statement-1 : ` sum_(r=0)^(n) r^(2) ""^(n)C_(r) x^(r) = n (n-1) x^(2) (1 + x)^(n-2) + nx (1 +x)^(n-1)`Statement-2: `sum_(r=0)^(n) r^(2) ""^(n)C_(r) = n (n-1)2^(n-2)+ n2^(n-1)` .A. 1B. 2C. 3D. 4 |
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Answer» Correct Answer - b We have, ` sum_(r=0)^(n) r^(2) ""^(n)C_(r) x^(r)` ` = sum_(r=0)^(n) {r( r-1)+r}""^(n)C_(r)x^(r)` `= sum_(r=0)^(n) r(r -1) ""^(n)C_(r) x^(r) sum_(r=0)^(n) r ""^(n)C_(r) x^(r)` `sum_(r=0)^(n) r(r -1) xx(n)/(r)xx(n-1)/(r-1) ""^(n-2)C_(r-2) x^(r) -2x^(2) + sum_(r=1)^(n)r xx (n)/(r) xx""^(n-1)C_(r-1) x^(r-1) xxx` ` = n (n-1) x^(2) sum_(r=2)^(n) ""^(n-1)C_(r-2) x^(r-2) + nx sum_(r=1)^(n) ""^(n-1)C_(r-1) x^(r-1)` `= n(n-1) x^(2) (1 + x)^(n-2) +nx (1 + x)^(n-1)` So, statement-2 s also true. Clearly, statement-1 is a correct expanation for statememt -2. But, statement-2 is not a correct expanation for statement-1. |
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