Question

Coefficient of $x^9$ in $\to (1+x)\left(\right(1+x^2\left)\right(1+x^3)......(1+x^{100}\left)\right)?$

Answer 1

Coefficient of $x^9$

$(1+x)(1+x^2)(1+x^3)\dots \dots ..(1+x^{100})$

First observe that $x^9$ can be only generate by products of the form.

$1.x^9,x^2.x^7,x^4.x^3.x^2$, $x^4.x^3.x^2$ and so on.

Therefore all we need to find is the number of ways the integer g can be partitioned into integers from the set.

$\{1,2,3,4,5,6,7,8,9\}$ These are

$\left\{9\right\},\left\{1.8\right\}$, $\{1,2,6\},\{1,3,5\},\{2,7\},\{3,6\},\{4,5\}$

$\{2,3,4\}$

There are $8$ such terms. Therefore the coefficient of $x^9$ is $8$.