If \({1}^{2}\cdot \left({}^{15}\mathrm{C}_{1}\right)+{2}^{2}\cdot \left({}^{15}\mathrm{C}_{2}\right)+{3}^{2}\cdot \left(…

If \({1}^{2}\cdot \left({}^{15}\mathrm{C}_{1}\right)+{2}^{2}\cdot \left({}^{15}\mathrm{C}_{2}\right)+{3}^{2}\cdot \left({}^{15}\mathrm{C}_{3}\right)+\ldots .+{15}^{2}\cdot \left({}^{15}\mathrm{C}_{15}\right)=\)\({2}^{\mathrm{m}}\cdot {3}^{\mathrm{n}}\cdot {5}^{\mathrm{k}}\), where \(\mathrm{m},\mathrm{n},\mathrm{k}\in \mathrm{N}\), then \(\mathrm{m}+\mathrm{n}+\mathrm{k}\) is equal to :-

[JEE Main 2025, 4 Apr (Shift 2)]