Browse Questions

# The value of $^nC_0.^{2n}C_r$ - $^nC_1^{2-2}C_r$ + $^nC_2.^{2n-4}C_r$ - ...., if $r \lt n$ is

$\begin{array}{1 1} (A) 2^{2n-r}.^nC_{r-n}\\ (B) 2^{2n-r+2}.^nC_{r-n+1}\\ (C) 2^{2n-r}.^nC_{r} \\ (D) 0 \end{array}$

$\big[(1+x)^2-1\big]^n=^nC_0.(1+x)^{2n}-^nC_1.(1+x)^{2n-2}+^nC_2.(1+x)^{2n-4}-.............$
$\Rightarrow\:x^n(2+x)^n=^nC_0.(1+x)^{2n}-^nC_1.(1+x)^{2n-2}+^nC_2.(1+x)^{2n-4}-.............$
Equating the coefficient of $x^r$ on both the sides,
Coeff. of $x^r\:\: in\:\: x^n(2+x)^n=$ coeff. of $x^{r-n}$ in $(2+x)^n$