So in one of the combinatorics book I am reading, it says:
$$(1+x_1)(1+x_2)...(1+x_n)=\sum_{T\subseteq S} \prod_{x_i\in T} x_i$$
Where $x_1,x_2,...,x_n$ are the elements of the $n$ set $S$, and $T$ is any subset of $S$. My question is, why is the right hand side of the equality the way it is? Because according to the left side, if you expand there should be the terms with $x_i$'s and a $1$, however on the rightside if you expand you dont get a term with $1$, so I dont understand how this equality holds. Can anyone help me clarify my confusion?
It may be a convention where if $T = \emptyset$ then $$\prod_{x_i\in T}x_i=1.$$
See the wikipedia page on empty product: https://en.wikipedia.org/wiki/Empty_product