# Thread: An easy problem!

1. ## An easy problem!

Let $M(n)=\{-1,\dots, -n\}$. Define the empty product to be $1$. For every subset of $M(n)$, multiply its elements together and add up the resulting $2^n$ numbers; what is the result?

2. Spoiler:
Given a subset $S$ of $M(n),$ define its twin subset $S^*$ to be $S^*=S\cup\{-1\}$ if $-1\notin S$ and $S^*=S\setminus\{-1\}$ if $-1\in S.$ Then it is clear that the twin subset of $S^*$ is $S$ itself, that all the $2^n$ subsets of $M(n)$ (provided $n\ne0)$ can be partitioned into $2^{n-1}$ pairs of mutually twin subsets, and that if the product of all the elements of $S$ is $k$ then the product of all the elements of $S^*$ is $-k.$ Hence the answer to the problem is $0.$

3. Good!

Here's my solution :

Spoiler:

Expand $0=(1-1)(1-2)(1-3)\dots(1-n)$.

4. Thanks Bruno.