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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.

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

2 Answers 2

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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 $(2^{90}-1)^{100}$.

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  • $\begingroup$ This makes perfect sense and is way simpler than what I was attempting. Thanks a bunch! $\endgroup$
    – dandan
    Dec 8, 2020 at 9:57
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Look at the power set of the set of all options - that would have $2^{90} - 1$ elements (removing the empty set)

Now, each person can choose any one of these options, hence the total number of ways would be

$$(2^{90}-1)^{100}$$

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