convert product to summation

Question

I would like to write the following function as a summation

$F(x) = \prod_{n=1}^{N} (1-e^{-\frac{x}{\gamma_{n}}})$.

I could not figure out how to expand it? Can someone please give me a hint?

Thanks.

Answer 1

$$\prod_{n=1}^N (1+x_n) = \sum_{A \subseteq \{1,\dots,N\}} \prod_{a\in A} x_a$$