<s>[INST] Find the coefficient of $x^9$ in $(1+x)(1+x^2)(1+x^3)\cdots(1+x^{100})$ This question had come in jee advanced 2015. Give a hint to solve it.
 [/INST] You have $100$ factors, each a sum of a $1$ and a positive power of $x$.  When the product is expanded as a sum, every term results from picking a $1$ from some factors and a positive power of $x$ from others.  If the positive powers of $x$ that you pick in one case are $x^3$, $x^{20}$, and $x^{35}$, then the term is $x^{3+20+35} = x^{58}$.
If they are $x^2$, $x^3$, and $x^4$, then the term is $x^{2+3+4}=x^9$. ${}\qquad 2+3+4=9$.
If they are $x^1$, $x^2$, and $x^6$, then the term is $x^{1+2+6} = x^9$. ${}\qquad 1+2+6=9$.
So the question is: In how many ways can you write $9$ as a sum of distinct positive integers?  That will be the coefficient of $x^9$.
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