We would like to prove by induction that the number of subsets of cardinality $2$ of a finite set with $n$ elements is given by $\frac{n(n-1)}{2}$.
I know the reason why this is true, but how could I use it in the induction process? Something seems to be missing here.
Suppose the number of $2$-element subsets of a $k$-element set is $\frac{k(k-1)}{2}$. Consider the number of $2$-element subsets of a set with an additional element. Such a subset can be chosen from the original $k$ elements, or it can be formed from taking the additional element plus an element of the original set. Thus there are an additional $k$ such subsets. Hence the total number of $2$-element subsets is $\frac{k(k-1)}{2}+k=\frac{k(k+1)}{2}$.