<s>[INST] Coefficient of $x^9$ in the Product Find the Coefficient of $x^9$ in $$G(x)=(1+x)(1+x^2)(1+x^3)(1+x^4)\cdots(1+x^{100})$$
My Try:
$$G(x)=P(x)(1+x^{10})(1+x^{11})\cdots(1+x^{100})=P(x)(1+O(x^{10}))$$ Hence Coefficient of $x^9$ in $G(x)$ is Coefficient of $x^9$ in $P(x)$ where $$P(x)=(1+x)(1+x^2)(1+x^3)\cdots(1+x^9)=R(x)+x^9R(x)$$  So Coefficient of $x^9$ in $P(x)$ is $1$+coeff of $x^9$ in $ R(x)$ where $$R(x)=(1+x)(1+x^2)\cdots(1+x^8)$$
But the method is becoming lengthy. Can i have any good approach
 [/INST] HINT: The desired coefficient is the number of ways to express $9$ as a sum of distinct positive integers. Since $1+2+3+4>9$, you need only consider sums of at most three integers.
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