Is there a series expansion formula for $\prod_{j=1}^{n} (a_j - b_j)$?

Question

I would like to expand the product $\prod_{j=1}^{n} (a_j - b_j)$ into a summation. Since there is some $n$ number of factors, I cannot simply 'foil' out the expansion. Checking for any specific value of $n$ by hand, there always seems to such a polynomial so I am simply stuck on navigating Pi-notation.

Is there a series expansion formula for $\prod_{j=1}^{n} (a_j - b_j)$?

Answer 1

$$ \prod_{j=1}^n (a_j - b_j) = \sum_{S \subseteq \{1,\ldots,n\}} \prod_{j \in S} a_j \prod_{j \in \{1,\ldots,n\} \backslash S} (-b_j)$$

That is, there is a term in the sum for every subset $S$ of the indices $1, \ldots, n$. That term consists of the product of $a_j$ for $j$ in that subset, times the product of $-b_j$ for $j$ not in that subset.