# An easy problem!

• Mar 19th 2010, 09:39 AM
Bruno J.
An easy problem!
Let $\displaystyle M(n)=\{-1,\dots, -n\}$. Define the empty product to be $\displaystyle 1$. For every subset of $\displaystyle M(n)$, multiply its elements together and add up the resulting $\displaystyle 2^n$ numbers; what is the result?
• Mar 19th 2010, 01:38 PM
proscientia
Spoiler:
Given a subset $\displaystyle S$ of $\displaystyle M(n),$ define its twin subset $\displaystyle S^*$ to be $\displaystyle S^*=S\cup\{-1\}$ if $\displaystyle -1\notin S$ and $\displaystyle S^*=S\setminus\{-1\}$ if $\displaystyle -1\in S.$ Then it is clear that the twin subset of $\displaystyle S^*$ is $\displaystyle S$ itself, that all the $\displaystyle 2^n$ subsets of $\displaystyle M(n)$ (provided $\displaystyle n\ne0)$ can be partitioned into $\displaystyle 2^{n-1}$ pairs of mutually twin subsets, and that if the product of all the elements of $\displaystyle S$ is $\displaystyle k$ then the product of all the elements of $\displaystyle S^*$ is $\displaystyle -k.$ Hence the answer to the problem is $\displaystyle 0.$
• Mar 19th 2010, 03:05 PM
Bruno J.
Good!

Here's my solution :

Spoiler:

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