Coefficient of an expansion

Question

Find the coefficient of $x^k$ in $(x+a)(x+b)...(x+n)$ where $a$, $b$ and $n$ are integers.

I am not able to approach this problem.

Answer 1

Expanding the polynomial will result in:

$$(x+a)(x+b)...(x+n)=x^n + (a + b + ... + n)x^{n-1} + (ab + ac + ... + an + ... + bn + ...)x^{n-2} + ...$$

So the coefficient of $x^k$ will simply be $$\sum_{a_1, a_2, ... a_{n-k}} a_1 a_2 ... a_{n-k},$$

where $a_1, a_2, ..., a_{n-k} \in {a, b, ..., n}$ and $a_i \neq a_j$ for every $i \neq j$.

In other words this is called a symmetric polynomial, so what you're looking for is $e_k(a, b, ..., n)$.