A combinatorial identity for the p-binomial coefficient based on abelian groups

For non-negative integers $k\leq n$, we prove a combinatorial identity for the $p$-binomial coefficient $\binom{n}{k}_p$ based on abelian p-groups. A purely proof of this is not known. While proving identity, $r\in \mathbb{N}\cup\{0\},s\in \mathbb{N}$ and $p$ prime, present formula number subgroups $\mathbb{Z}^s$ finite index $p^r$ with quotient isomorphic to $p$-group type $\underline{\lambda}$, which partition $r$ into at most $s$ parts. This similar that enumeration certain in obtained by Lynne Marie Butler. As consequences, gives rise many formulae involve polynomials integer coefficients.