The coefficient of $x^{9}$ in the expansion of $(1+x)\left(1+x^{2}\right)$ $\left(1+x^{3}\right) \ld

Question

The coefficient of $x^{9}$ in the expansion of $(1+x)\left(1+x^{2}\right)$ $\left(1+x^{3}\right) \ldots\left(1+x^{100}\right)$ is

Professor PC Thomas Classes · Answer

(8) : $x^{9}$ can be obtained as a sum of two numbers in 5 ways, i.e., $(0,9),(1,8)$, $(2,7),(3,6),(4,5)$ and as a sum of three numbers in 3 groups, i.e., $(1,2,6),(1,3,5)$, (2, 3, 4).\ Thus the total number of ways $=8$\