Let $m \leq n$ be positive integers and $p$ be a prime. Let $p-$expansions of $m$ and $n$ be \[m = a_0 + a_1p + \dots + a_rp^r\]\[n = b_0 + b_1p + \dots + b_sp^s\] respectively, where $a_r, b_s \neq 0$, for all $i \in \{0,1,\dots,r\}$ and for all $j \in \{0,1,\dots,s\}$, we have $0 \leq a_i, b_j \leq p-1$ . If $a_i \leq b_i$ for all $i \in \{0,1,\dots,r\}$, we write $ m \prec_p n$. Prove that \[p \nmid {{n}\choose{m}} \Leftrightarrow m \prec_p n\].