Coefficient of $x^9$ in $(1+x)\cdot (1+x^2)\cdot (1+x^3)\cdots(1+x^9)$ 
The Coefficient of $x^9$ in $(1+x)\cdot (1+x^2)\cdot (1+x^3)\cdots(1+x^9)$ is 

My Try We Know that Coefficient  of $x^9$ occur (which is $ = 1$) when we 
multiply $x^0\cdot x^9$ and $x^1\cdot x^8$ and $x^2\cdot x^7$ and $x^3\cdot x^6$ and $x^4\cdot x^5$
Now I did not understand How can I calculate it bcz answer given as $ = 8$
Help me, Thanks 

This is the same as the number of ways to express $9$ as the sum of distinct positive integers $\le 9$.
$9 = 9 = 8+1 = 7+2 = 6+3 = 6+2+1 = 5+4 = 5+3+1 = 4+3+2$
So there are eight ways. 
