If $1^{2}\cdot{^{15}C_{1}}+2^{2}\cdot{^{15}C_{2}}+3^{2}\cdot{^{15}C_{3}}+\cdots+15^{2}\cdot{^{15}C_{15}}=2^{m}\cdot3^{n}\cdot5^{k}$, where $m,n,k\in\mathbb{N}$, then $m+n+k$ is equal to: