Question

Prove that in the expansion of `(1+a)^(m+n)`, coefficients of `a^m` and `a^n` are equal?

Answer 1

Please see below.

Explanation:

The expansion of `(1+x)^n=C_0^n+C_1^nx+C_2^nx^2+.....+C_r^nx^r+....+C_n^nx^n`,

where `C_r^n=(n!)/(r!(n-r)!)`

It is apparent that in the expansion of `(1+a)^(m+n)`,

coefficient of `a^m` is `C_m^(m+n)=((m+n)!)/(m!(m+n-m)!)=((m+n)!)/(m!n!)`

and coefficient of `a^n` is `C_n^(m+n)=((m+n)!)/(n!(m+n-n)!)=((m+n)!)/(n!m!)`

As such in the expansion of `(1+a)^(m+n)`, coefficients of `a^m` and `a^n` are equal.