First please note that a 2-subset is a subset of a set on n elements containing exactly 2 elements, so the number of 2-subsets of an n-set is just . Now let's prove it by induction:

Base Step: When , it is trivial (we have 2-subset); when , we have 2-subset.

Inductive Step: Now lets assume when , we have 2-subsets. Consider the case when (i.e. we add one more element into the set, so how manymore2-subsets will be introduced? The answer ismore2-subsets are introduced. Think about this)

So the total number of 2-subsets when is , this competes the inductive step.

Roy