# 100 people choose any number of combinations from a pool of 90 options.

If 100 people can choose any number of combinations from a set of 90 options, where each option can only be selected once, but each person can select any number of options.

Say there is a product sortiment of 90 different products, [a, b, c...]

Each person can choose any combination of products, but can only choose each product once. So for example a person can choose [a] or [a, b, f, g] but not [a, a, c].

Now if 100 people has to make a selection and each selection doesn't affect other peoples selections. How many combinations does this equal?

EDIT: I forgot to add: [a, b, c] and [b, a, c] should count as the same combination.

• Can a person choose zero options? Can multiple people choose the same option? Do all options have to be chosen by some person? Dec 8, 2020 at 9:44
• A person can't choose zero options. Multiple people can choose the same option. All options doesn't have to be chosen. Dec 8, 2020 at 9:50
• If two persons interchange their options is it a distinct combination?
– user
Dec 8, 2020 at 9:51
• @user I am not sure what you mean by this? :-) Dec 8, 2020 at 9:52
• If the person 1 have chosen a and the person 2 have chosen b is it the same as person 1 have chosen b and the person 2 have chosen a.
– user
Dec 8, 2020 at 9:54

Each person individually may or may not choose any option, but has to choose at least one option. Thus each person may choose in $$2^{90}-1$$ ways. Since the choices are independent between people, the answer is $$(2^{90}-1)^{100}$$.
Look at the power set of the set of all options - that would have $$2^{90} - 1$$ elements (removing the empty set)
$$(2^{90}-1)^{100}$$