Given a set s how many subsets of s contain at most 2 elements

Given a set S of cardinality , the subsets containing at most 2 elements are:

There is clearly only one empty set, and there are subsets containing one element (you choose one of the n elements in S to create your subset)

Finally, the number of subsets containing two elements is

In fact, in general, the binomial coefficient represents the number of subset of cardinality k that you can build from a set with cardinality n.

So, if we update our list, we have the following subsets:

For a total of

subsets with at most two elements.