If A and B are the coefficient of


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 $\frac{A}{B}=$ ________________


The coefficient of xn is (1 + x)2n is ?

Since Tr +1 = 2nCr xr

For xcoefficient put r = n

i.e coefficient of xis 2nC

i.e. A = 2nCn

and for coefficient of xin (1 + x)2n–1

Tr +1 = 2n–1Cr xr

Put r = n

i.e. coefficient of xin (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

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