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# $^nC_0.^nC_1$ + $^nC_1.^nC_2$ + $....^nC_{n-1}.^nC_n$ = ?

$\begin{array}{1 1}\large\frac{(2n)!}{(n-1)!.(n+1)!} \\ \large\frac{(2n)!}{(n)!.(n+1)!} \\ \large\frac{(2n)!}{(n-1)!.(n)!} \\ \large\frac{(2n)!}{(n-1)!.(n-1)!}\end{array}$

$(1+x)^n=^nC_0+^nC_1.x+^nC_2.x^2+........^nC_nx^n$........(i)
$(x+1)^n=^nC_0x^n+^nC_1.x^{n-1}+^nC_2.x^{n-2}+........^nC_n$....(ii)
comparing the coefficient of $x^{n-1}$ on both the sides in the product of (i) and (ii) we get
$\large\frac{(2n)!}{(n-1)!.(n+1)!}$= $^nC_0.^nC_1+^nC_1.^nC_2+^nC_2.^nC_3+............^nC_{n-1}.^nC_n$
edited Oct 28, 2013