Question

Coefficient of $x^{n-2}$ in expansion of $(x-1)(x-2)(x-3)....(x-n)$

Answer 1

$(x-1)(x-2)$____ $(x-n)$

For coefficient of $x^{n-2},2$ terms will be left out

$\Rightarrow$ Coefficient $=1.2+2.3+3.4+\dots .+(n-1)n$

$=\sum _{r=1}^{n-1}r(r+1)=\sum _{r=1}^{n-1}{r}^2+r$

$=\frac{(n-1)(n+1)\left(2\right(n-1)+1)}{6}+\frac{(n-1)n}{2}$

$=\frac{n(n-1)(2x-1)}{6}+\frac{n(n-1)}{2}$

$=\frac{n(n-1)}{6}\left[\right(2n-1)+3]$

$=\frac{2n(n-1)(n+1)}{6}=\frac{n(n-1)(n+1)}{3}$