Suppose $x^n$ occurs in the (r + 1)th term of the expansion $(1 + x)^{2n}$. Now $T_{r+1}$ = $\,{}^{2n}C_r x^r$ On comparing power of x in $x^n$ and $T_{r + 1}$, we get r = n Thus the coefficient of $x^n$ is $\,{}^{2n}C_n$ = $\frac{2n!}{n!n!}$ = $\frac{2n(2n-1)!}{n!n(n-1)!}$ = $\frac{2(2n-1)!}{n!(n-1)!}$ ... (i) Now suppose $x^n$ occurs in the (r + 1)th term of the expansion $(1+x)^{2n-1}$ Now $T_{r+1}$ = $\,{}^{2n-1}C_r x^r$ On comparing power of x in $x^n$ and $T_{r + 1}$, we get r = n Thus the coefficients of $x^n$ = $\,{}^{2n-1}C_n$ = $\frac{(2n-1)!}{n!(n-1)!}$ ... (ii) From (i) & (ii), we see that coefficient of $x^n$ in the expansion of $(1 + x)^{2n}$ is twice the coefficient of $x^n$ in the expansion of $(1 + x)^{2n - 1}$. Hence proved.