A person has 10 friends and over several days invites some of them to visit so that the company never repeats (on some days he may not invite anyone). How many days can he do this?

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The first method. For each of the 10 friends, there are two possibilities: to invite them or not. Therefore, the total number of ways to choose a group of friends is $2^{10}$.

The second method. The problem is equivalent to finding the number of subsets of a set with 10 elements. As is known, there are $2^{10}$ of them.

## Answer

1024 days.