Doubt in finding number of integral solutions Problem : 
writing $5$ as a sum of at least $2$ positive integers. 
Approach :
I am trying to find the coefficient of $x^5$ in the expansion of $(x+x^2+x^3\cdots)^2\cdot(1+x+x^2+x^3+\cdots)^3$ .
which reduces to coefficient of $x^3$ in expansion of $(1-x)^{-5}$ ,which is $${7\choose3}= 35$$
but we can count the cases and say that answer must be $6$ :
$4 + 1 $
$3 + 2 $
$3 + 1 + 1$
$2 + 2 + 1 $
$2 + 1 + 1 + 1$
$1 + 1 + 1 + 1 + 1$
At which stage am I making a mistake ? Thanks

There are $2^{5-1} -2^{(2-1)-1}=15$ compositions of $5$ into at least $2$ positive integers while there are $6$ partitions of $5$ into at least $2$ positive integers.  With compositions, $1+1+3$ is distinct from $3+1+1$, while with partitions they are the same.
But your first attempt counts even more than compositions.  It actually counts cases where the first two elements are positive and the next three are non-negative, including $3+1+0+0+1$.  I doubt this is what you want to count.
