InterviewSolution
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InterviewSolution
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If `C_(0), C_(1), C_(2), ..., C_(n)` denote the binomial cefficients in the expansion of `(1 + x )^(n)` , then`1^(2).C_(1) + 2^(2) + 3^(3).C_(3) + ...+n^(2).C_(n)=`.A. `(n + 1)2^(n-2)`B. `n(n + 1)2^(n-1)`C. `n(n + 1)2^(n-2)`D. `n(n-1) 2^(n-2)` |
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Answer» Correct Answer - c We have, `1^(2).C_(1) + 2^(2) + 3^(3).C_(3) + ...+n^(2).C_(n)` `= sum_(r=1)^(n) r^(2) .C_(r)` `= sum_(r=1)^(n) r^(2) .""^(n)C_(r)` `= sum_(r=1)^(n) r(r-1) ""^(n)C_(r)+sum_(r=1)^(n) r.""^(n)C_(r)` `= sum_(r=1)^(n) r(r-1) .(n)/(r).(n-1)/(r-1) ""^(n-2)C_(r-2)+sum_(r=1)^(n) r.(n)/(r)""^(n-1)C_(r-1)` ` = n(n -1) ( sum_(r=1)^(n) ""^(n-2)C_(r-2))+n(sum_(r=1)^(n) ""^(n-1)C_(r-1))` `n(n-1)(""^(n-2)C_(0) + ""^(n-2)C_(1) + ...+""^(n-2)C_(n-2)}` `n+{""^(n-1)C_(0) + ""^(n-1)C_(1) + ...+""^(n-1)C_(n-1)}` `n(n-1).2^(n-2) + n^(n-1)` `= n(n-1+2).2^(n-2) = n(n+1)2^(n-2)` |
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