I only have 32 bits of space to store a combination of items. The constraints are that a combination can have anywhere from 0 to 7 items. There are 64 unique items. An item cannot appear multiple times in a combination.
How do I map these sets of items to 32-bit integers?
I have to assume that by a "combination" you mean a set. The ordering of a set does not matter. If the ordering matters, then the possibilities cannot fit in 32 bits.
For set of 0 to 7 elements, where there are 64 possible elements, we have that there are the sum of binomial(64,i) for i in 0..7 possibilities, which is 704,494,193. That would even fit in 30 bits (230 = 1,073,741,824).
Simply define a scheme that numbers each possibility uniquely and store that number. E.g. assign 0 for zero elements, 1..64 for one element, 65..2080 for two elements, and so on.