Ii don't know share your approach!
Actually, I need the ordered collection only ..sorry for describing the QS as an unordered set. @kapil asked earlier about this.
will this recurrence work ? for ordered collection?
f(m,k)
= no of ordered collection using the numbers from S
$f(m,k) = \begin{cases} 1 \qquad m=0 , k = 0\\ \\ 0 \qquad m=0 , k > 0\\ \\ \sum_{x_i \; }^{\{\{0\} \cup S \}\;} f(m-1,k-x_i) \quad m \geq 1 , (k+x_i) \geq 0 \\ \end{cases}$