General formula for the number of terms from when evaluating the product of n terms $\Pi_{i=1}^{n}(1+x_i)^n$?

Question

I'm wondering whether there is a formula for the number of terms in the expression $\Pi_{i=1}^{n}(1+x_i)^n$ when multiplying all the terms out. I'm postulating that the number of terms is equal to $2^n$ but how to prove it?

Answer 1

$(1+x_i)^n$ will have $n+1$ terms, by the binomial theorem, and so multiplying indexed over $i$ will give $(n+1)^n$ total terms since each term will uniquely come from one of the $n$ multiplicands, each multiplicand giving $n+1$ choices.

Answer 2

Notice first that $(1+x_i)^n$ contains $n+1$ so if you do not do anything but multiplying them you will get $(n+1)^n$ but if you put them together, notice that you have $n$ variables and each variable either appears or it does not appear in every monomial. There are $2$ choices for each of the $n$ variables, then you have $$\underbrace{2\times 2\cdots \times 2\times 2}_{\text{n times}}=2^n$$ as claimed.