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# $^nC_0^2$ + $^nC_1^2$ + $^nC_2^2$ + ...... + $^nC_n^2$ = ?

$\begin{array}{1 1} n.^{2n}C_n \\ (n+1).^nC_n^2 \\ n.^nC_2^2 \\ ^{2n}C_n \end{array}$

$(1+x)^n=^nC_0+^nC_1x+^nC_2x^2+............^nC_nx^n$....(i)
$(x+1)^n=^nC_0x^n+^nC_1x{n-1}+^nC_2x^{n-2}+........^nC_n$...(ii)
$(i)\:\times\:(ii)=(1+x)^{2n}$
Comparing the coeff. of $x^n$ in the product of $(i)\:and\:(ii)$ on either sides,
coeff. of $x^n$ in the expansion of $(1+x)^{2n}=^nC_0^2+^nC_1^2+^nC_2^2+....^nC_n^2$
$\Rightarrow\:^nC_0^2+^nC_1^2+^nC_2^2+....^nC_n^2=^{2n}C_n$