Mathematics - Binomial Theorem Question with Solution | TestHub

$(x-1)(x-2)(x-3)\dots ..(x-n)$ $=x^n-\left({\sum }_{i=1}^{n}i\right)x^{n-1}+\left({\sum }_{1\le i<j\le n}i\cdot j\right)x^{n-2}-\dots .$ Coefficient of $x^{n-2}={\sum }_{i<j}i.j=\frac{{\left(\sum i\right)}^2-\sum i^2}{2}=\frac{1}{2}\left[\frac{n^2(n+1{)}^2}{4}-\frac{n(n+1)(2n+1)}{6}\right]$ $=\frac{n(n+1)\left(3n^2-n-2\right)}{24}=\frac{n(n+1)(n-1)(3n+2)}{24}=\frac{n\left(n^2-1\right)(3n+2)}{24}$