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)`

Explanation

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.