<s>[INST] Coefficient of $x^9$ 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?
 [/INST] 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:
$9, 1+8, 2+7, 3+6, 4+5, 1+2+6, 1+3+5, 2+3+4$
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