How to find the coefficient of $x^8$ in $\prod\limits_{i=1}^{10}{\left(x-i\right)}$? How to find the coefficient of $x^8$ in $(x-1) (x-2)  . . .(x-10)$.
Is there any general formula to solve this kind of problems?

The roots of the polynomial $(x - 1)(x-2)\cdots(x-10) = 0$ are $x = 1,2,\dots,10$.
Apply Vieta's Formulas to deduce that the coefficient of $x^8$ is the sum of all pairwise products of  these roots (a.k.a the second elementary symmetric function of the roots):
$$(1\cdot2 + 1\cdot 3 + \dots + 1\cdot 10) + (2\cdot 3 + 2\cdot 4 + \dots + 2\cdot 10) + \dots + (9\cdot 10)\\
= 1320$$
Hence, the coefficient of $x^8$ is $1320$. The Wolf verifies.
