Mathematics - Permutation & Combination Question with Solution | TestHub

$\begin{array}{rl}& {\left(x^1+x^2\dots +x^6\right)}^4\\ & x^4\cdot {\left(\frac{1-x^6}{1-x}\right)}^4\\ & x^4\cdot {\left(1-x^6\right)}^4\cdot (1-x{)}^{-4}\\ & x^4\left[1-4x^6+6x^{12}\dots .\right]\left[(1-x{)}^{-4}\right]\\ & \left(x^4-4x^{10}+6x^{16}\dots \right)(1-x{)}^{-4}\\ & \left(x^4-4x^{10}+6x^{16}\right)\left(1+{}^{15}{C}_{12}{x}^{12}+{}^9{C}_6{x}^6\dots .\right)\\ & \left({}^{15}{C}_{12}-4\cdot {}^9{C}_6+6\right)x^{16}\\ & \left({}^{15}{C}_3-4\cdot {}^9{C}_6+6\right)\\ & =35\times 13-6\times 8\times 7+6\\ & =455-336+6\\ & =125\end{array}$