InterviewSolution
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InterviewSolution
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If `C_(0),C_(1), C_(2),...,C_(N)` denote the binomial coefficients in the expansion of `(1 + x)^(n)` , then `1^(3). C_(1)-2^(3). C_(3) - 4^(3) . C_(4) + ...+ (-1)^(n-1)n^(3) C_(n)=` |
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Answer» Correct Answer - a We have, `1^(3). C_(1)-2^(3). C_(3) - 4^(3) . C_(4) + ...+ (-1)^(n-1)n^(3) C_(n)` `= sum +(r=1)^(n) (-1)^(r-1) r^(3) ""^(n)C_(r)` ` = sum_(r=1)^(n) (-1)^(r-1) {r(r -1) (r-2)+ 3r (r-1) + r} . ""^(n)C_(r)` ` = sum_(r=1)^(n) (-1)^(r-1) r(r -1) (r-2). ""^(n)C_(r)+ 3 sum_(r=1) ^(n) (-1)^(r-1)r(r-1) ""^(n)C_(r) + sum_(r-1)^(n) (-1)^(r-1) r. ""^(n)C_(r)` `sum_(r=1)^(n) (-1)^(r-1)r(r-1) (r-2).(n)/(r).(n-1)/(r-2).(n-2)/(r-2)""^(n-3)C_(r-3)` `+3sum_(r=1)^(n) (-1)^(r-1)r.(n)/(r) ""^(n-1 )C_(r-1)` `= n(n -1) (n-2) {sum_(r=1)^(n) (-1)^(r-3) ""^(n-3)C_(r -3)}` ` - 3 (n-1) sum _(r=1)^(n) (-1)^(r -2)""^(n-2)C_(r-2+ n) {sum_(r=1)^(n) (-1)^(r -1)""^(n-1)C_(r-1)}` `=n(n-1) (n-2)xx0 - 3n(n-1)xx0+n xx- = 0` |
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