Question:
If $A$ and $B$ are the coefficient of $x^{n}$ in the expansion of $(1+x)^{2 n}$ and $(1+x)^{2 n-1}$ respectively, then $\frac{A}{B}=$ ________________
Solution:
The coefficient of xn is (1 + x)2n is ?
Since Tr +1 = 2nCr xr
For xn coefficient put r = n
i.e coefficient of xn is 2nCn
i.e. A = 2nCn
and for coefficient of xn in (1 + x)2n–1
Tr +1 = 2n–1Cr xr
Put r = n
i.e. coefficient of xn in (1 + x)2n–1 is 2n–1Cn
i.e B = 2n–1Cn
$\therefore \frac{A}{B}=\frac{{ }^{2 n} C_{n}}{{ }^{2 n-1} C_{n}}=\frac{(2 n) ! n ! n !}{n ! n !(2 n-1) !}$
= 2