InterviewSolution
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InterviewSolution
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If `""(n)C_(0), ""(n)C_(1), ""(n)C_(2), ...., ""(n)C_(n), ` denote the binomial coefficients in the expansion of `(1 + x)^(n) and p + q =1` ` sum_(r=0)^(n) r^(2 " "^n)C_(r) p^(r) q^(n-r) = ` .A. npqB. np (p+q)C. `np (np + q)`D. ` np (p + nq)` |
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Answer» Correct Answer - b We have, ` sum _(r=0)^(n) r^(2)""^(n)C_(r)p^(r) q^(n-r)` ` sum _(r=0)^(n) [r (r-1)+""^(n)C_(r)p^(r) q^(n-r)` ` sum _(r=0)^(n) r (r-1)""^(n)C_(r)p^(r) q^(n-r) +sum_(r-0)^(n) r. ""^(n)C_(r) p^(r) q^(n-r)` = `n(n - 1){sum_(r=0)^(n) ""^(n-2)C_(r-2) p^(r)q^(n-r) }+n{ sum_(r=0)^(n)""^(n-1)C_(r-1) p^(r)q^(n-r) } ` `= n(n -1) P^(2)` = `n(n - 1){sum_(r=0)^(n) ""^(n-2)C_(r-2) p^(r-2)q^((n-2)-(r-2))}` `+ np {sum_(r=0)^(n) ""^(n-1) C_(r-1) p^(r -1) q^((n-1)-(r-1))}` `= n(n-1) p^(2) (p+q)^(n-2) + np (p + q)^(n -1)` =` n(n -1) p^(2) + np [because p + q = 1]` ` n^(2) p^(2) - np^(2) + np` ` = n^(2) p^(2) + (1 - P ) = n^(2) p^(2) + np q [ because p + q = 1]` |
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