Browse Questions

# The coefficient of $x^n$ in the expansion of $(1+x)^{2n}$ and $(1+x)^{2n-1}$ are in the ratio

$\begin{array}{1 1}(A)\;1 : 2\\(B)\;1 : 3\\(C)\;3 : 1\\(D)\;2 : 1\end{array}$

Toolbox:
• $T_{r+1}=nC_r a^{n-r} b^r$
$(1+x)^{2n}$
$T_{r+1}=2nC_n(1)^{2n-r}x^{2n}$
$(1+x)^{2n-1}$
$T_{r+1}=2n-1C_n(1)^{2n-1-r}x^{2n-1}$
The coefficient of $x^n$ is $2nC_n=2n-1C_n$
$\large\frac{2nC_n}{2n-1C_n}=\frac{2n!}{n!n!}.\frac{n!(n-1)!}{(2n-1)!}$
$\Rightarrow \large\frac{2n}{n}$$=2$
$\therefore 2nC_n : 2n-1C_n=2 : 1$
Hence (D) is the correct answer.