InterviewSolution
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The value of .^(n)C_(1)+.^(n+1)C_(2)+.^(n+2)C_(3)+"….."+.^(n+m-1)C_(m) is equal to |
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Answer» `.^(m+n)C_(n) - 1` `= .^(n)C_(n-1)+.^(n+1)C_(n-1)+.^(n+2)C_(n-1)+"……"+.^(n+m-1)C_(n-1)` `=` Coefficient of `x^(n-1)` in `(1+x)^(n) [((1+x)^(m) -1)/((1+x) - 1)]` = Coefficient of `x^(n-1)` in `((1+x)^(m+n) -(1+x)^(n))/(x)` `=` Coefficientof `x^(n)` in `[(1+x)^(m+n) - (1+x)^(n)]` `= .^(m+n)C_(n) - 1` SIMILARLY, we can prove `.^(m)C_(1)+.^(m+1)C_(2)+.^(m+2)C_(3)+"...."+.^(m+n-1)C_(n)=.^(m+n)C_(m)-1` |
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