Number of Selections of 1 or More from Set

Theorem
Let $S$ be a set of $n$ objects.

The total number $N$ of ways that at least $1$ object can be selected from $S$ is:
 * $N = 2^n - 1$

Proof
This is equivalent to counting the number of non-empty subsets of $S$.

From Cardinality of Power Set of Finite Set, the total number of subsets of $S$ is $2^n$.

This includes the empty set.

Excluding the empty set from the count gives the result.