# An easy problem!

• March 19th 2010, 09:39 AM
Bruno J.
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?
• March 19th 2010, 01:38 PM
proscientia
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 $0.$
• March 19th 2010, 03:05 PM
Bruno J.
Good!

Here's my solution :

Spoiler:

Expand $0=(1-1)(1-2)(1-3)\dots(1-n)$.
• March 20th 2010, 07:55 AM
funnyname7
Thanks Bruno.