Number of Selections from n Objects of 2 Types

Theorem
Let $S$ be a collection of $n$ objects, consisting of:
 * $p$ objects of one type
 * $q$ objects of another type.

The total number $N$ of different non-empty selections from $S$ is given by:
 * $N = 2^n \paren {p + 1} \paren {q + 1} - 1$

Proof
For each object, we have $2$ choices: whether to select or whether not.

From Cardinality of Power Set of Finite Set, this gives us $2^n$ options.

From the $p$ things of the first type, we may take $0, 1, 2, \ldots, p$ objects.

Hence we have $\paren {p + 1}$ choices.

Similarly, from the $q$ things of the second type, we have $\paren {q + 1}$ choices.

Hence we have $\paren {p + 1}$ choices.

Because the non-empty selection is excluded, this gives:
 * $N = 2^n \paren {p + 1} \paren {q + 1} - 1$

Hence the result.