InterviewSolution
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InterviewSolution
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Let m,`in` N and `C_(r) = ""^(n)C_(r)`, for ` 0 le r len` Statement-1: `(1)/(m!)C_(0) + (n)/((m +1)!) C_(1) + (n(n-1))/((m +2)!) C_(2) +… + (n(n-1)(n-2)….2.1)/((m+n)!) C_(n)` ` = ((m + n + 1 )(m+n +2)…(m +2n))/((m +n)!)` Statement-2: For r `le`0 `""^(m)C_(r)""^(n)C_(0)+""^(m)C_(r-1)""^(n)C_(1) + ""^(m)C_(r-2) ""^(n)C_(2) +...+ ""^(m)C_(0)""^(n)C_(r) = ""^(m+n)C_(r)`. |
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Answer» `1/(m!).^(n)C_(0)+(n)/((m+1)!).^(n)C_(1)+(n(n-1))/((m+2)!).^(n)C_(2)+"....."+(n(n-1)xx"...."xx2xx1)/((m+2)!).^(n)C_(n)` `= (n!)/((m+n)!)(((m+n)!)/(m!n!).^(n)C_(0)+((m+n)!n)/((m+1)!n!).^(n)C_(1)+((m+n)!n(n-1))/((m+2)!n!) xx .^(n)C_(2)+"....."+((m+n)!)/(n!)+"....."+((m+n)!)/(n!)(n(n-1)xx"....."xx2xx1)/((m+n)!).^(n)C_(n))` `= (n!)/((m+n)!)(.^(m+n)C_(n).^(n)C_(0)+((m+n)!)/((m+1)!(n-1)!).^(n)C_(1)+((m+n)!)/((m+2)!(n-2)!)xx.^(n)C_(2) + "....."+((m+n)!)/(1)(1)/((m+n)!)(1)/((m+n)!) .^(n)C_(n))`. `= (n!)/((m+n)!)(.^(m+n)C_(n).^(n)C_(0)+.^(m+n)C_(n-1).^(n)C_(1)+.^(m+n)C_(n-2).^(n)C_(2)+"...."+.^(m+n)C_(0).^(n)C_(n))` `= (n!)/((m+n)!)`[coefficient of `x^(n)` in `(1+x)^(m+n)(1+x)^(n)`] `= (n!)/((m+n)!)`[coefficient of `x^(n)` in `(1+x)^(m+2n)`] `= (n!)/((m+n)!).^(m+2n)C_(n)` `= (n!)/((m+n)!)((m+2n)!)/((m+n)!n!)` `= ((m+2n)!)/((m+n)!(m+n)!)` `= ((m+n+1)(m+n+2)(m+n+3)"...."(m+2n))/((m+n)!)` |
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