What does this mean exactly?

Question

I am having trouble understanding the following notation, which I encountered studying combinatorics:

\begin{equation} \sum_{A \subseteq \left[ n \right]} \prod_{a\in A}x_a \end{equation}

where $[n]:=\lbrace{1,\ldots,n\rbrace}$. Can someone please give me an intuitive explanation, or, perhaps better, a combinatorial one? Much thanks in advance.

Answer 1

It is a sum of products. For each subset of $[n]$ you multiply the corresponding $x$s. You then add all these products. For example, if $n=3$, we have

$$1+x_1+x_2+x_3+x_1 x_2+x_1 x_3+x_2 x_3+x_1 x_2 x_3 $$

Note the first term, $1$, corresponds to the term for the empty set, as the empty product is equal to $1$.

Answer 2

You have $n$ variables $x_1$ to $x_n$. Now form every product of some of these where each variable appears at most once. Add all these products together.