Counting subgroups of fixed order in finite abelian groups

We use recurrence relations to derive explicit formulas for counting the number of subgroups of given order (or index) in rank 3 finite abelian p-groups and use these to derive similar formulas in few cases for rank 4. As a consequence, we answer some questions by M. T$\ddot{a}$rn$\ddot{a}$uceanu in \cite{MT} and L. T$\dot{\acute{o}}$th in \cite{LT}. We also use other methods such as the method of fundamental group lattices introduced in \cite{MT} to derive a similar counting function in a special case of arbitrary rank finite abelian p-groups.