InterviewSolution
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InterviewSolution
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If m and n are positive integers, then prove that the coefficients of `x^(m) " and " x^(n)` are equal in the expansion of `(1+x)^(m+n)` |
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Answer» In the binomial expansion of `(1+x)^(m+n)` , the general term is given by ` T_(r+1) = ^(m+n)C_(r) xxx^(r)`. `:."coefficient of "x^(m) " in the expansion of "(1+x)^(m+n)` `.^(m+n)C_(m)=((m+n)!)/({(m+n-m)!}xx (m!))=((m+n)!)/((m!) xx(n !)).` coefficient of `x^(n)" in the expansion of "(1+x)^(m+n)` `=.^(m+n)C_(n)=((m+n)!)/({(m+n-n)!} xx(n!))+((m=n)!)/((m!) xx(n!)).` Hence, the coefficents fo `x^(m) and x^(n)` are equal. |
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