Question:
If $a$ and $b$ are the coefficients of $x^{n}$ in the expansion of $(1+x)^{2 n}$ and $(1+x)^{2 n-1}$ respectively, find $\frac{a}{b}$.
Solution:
Coefficients of $x^{n}$ in the expansion of $(1+x)^{2 n}$ is ${ }^{2 n} C_{n}=a$.
Coefficients of $x^{n}$ in the expansion of $(1+x)^{2 n-1}$ is ${ }^{2 n-1} C_{n}=b$.
Now,
$\frac{a}{b}=\frac{{ }^{2 n} C_{n}}{{ }^{2 n-1} C_{n}}$
$=\frac{\frac{(2 n) !}{n ! n !}}{\frac{(2 n-1) !}{n !(n-1) !}}$
$=\frac{2 n}{n}$
$=2$
Hence, $\frac{a}{b}=2$.
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