Q:

How to calculate unique combinations

Let's suppose that we have three variables: $xyz (n=3)$. We need to calculate how many unique combinations we can make. So in this case, you can simply get the answer without using any formulas: $xy, xz, yz, xyz$. So there are $4$ unique combinations. But how do you calculate it with some kind of formula when it gets more complicated? So for example, $4$ variables $wxyz$. Now you have $wx, wy, wz, xy, xz, yz, wxy, wxz, wyz, xyz, wxyz$ ($11$ combinations). And how do you do this when you have even $10$ variables? How do you do this?

A:

A set with $n$ elements has exactly $2^n$ subsets.  Here we do not want the empty set, nor do we want any set with exactly one element.  Thus the answer is $$2^n-n-1$$

