Coefficient of $x^9$

Question

Find the coefficient of $x^9$ in $(1+x)(1+x^2)(1+x^3)\cdots(1+x^{100})$.

Approach:-

By inspection, coefficient of $x^9$ will not be there in terms after $(1+x^{9})$

Now checking coefficient in $(1+x)(1+x^2)...(1+x^9)$.

Now by hit and trial, I checked what are the possible combinations of getting $x^9$.

I got the answer as $8$. Is there any other and better approach?

Answer 1

This question is equivalent to: How many sums of natural numbers equal $9$, subject to the condition that the terms be increasing? The terms in the sum are just exponents coming from all of the factors you mention.

For example, the sum $2+7=9$ corresponds to the term in your product where we take $x^2$, $x^7$, and $1$ from every other factor.

Thus, we list:

$9, 1+8, 2+7, 3+6, 4+5, 1+2+6, 1+3+5, 2+3+4$