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.

$\blacksquare$

Examples

$6$ Items

Let there be $6$ different flowers in a vase.

Then there are $63$ different ways of selecting at least one of these flowers.

Sources

• 1953: L. Harwood Clarke: A Note Book in Pure Mathematics ... (previous) ... (next): $\text I$. Algebra: Permutations and Combinations: The number of selections from $n$ things if any number may be taken