The coefficient of $x^9$ in the expansion of $(1+x)(1+x^2)(1+x^3)\cdots(1+x^{100})?$ What is the coefficient of $x^9$ in the expansion of $(1+x)(1+x^2)(1+x^3)\cdots (1+x^{100})?$

I manually expanded $(1+x)(1+x^2)(1+x^3)...(1+x^{10})$ and calculated the coefficient of $x^9$ as $8$ but i dont know how to solve it without expanding.Please help me.

In every factor of your product, you will need to pick either the $1$ or the $x^i$, so immediatly we see that for all factors with $i > 9$ we need to pick the $1$, otherwise we will get $x^k$ with $k > 9$. So we can just look at the product up till $(1 + x^9)$. Now we look at the number of ways to pick $1$ and $x^i$ here such that we end up with $x^9$. Because all our coefficients are $1$, we do not need to worry about those, and count only the number of ways.
To simplify this, assume you always pick the $1$ if you don't write them down, then all possible combinations are:
$x^9, x^1\cdot x^8, x^2\cdot x^7, x^3\cdot x^6, x^4\cdot x^5, x^1\cdot x^2\cdot x^6, x^1\cdot x^3\cdot x^5, x^2\cdot x^3\cdot x^4$ which gives us $8$ as the final result.
It is not hard to see that this is equivalent to summing numbers $1, \ldots 9$ to $9$ using every number only once, and then counting the number of summations.
