MathB.in α

Given $n$ object types, each with cost $n_{c}$, I need to combine object instances such that the total costs equal a known total $T$. Each object type can have $0 \le x \le y$ instances (where $x$ is a positive integer), but I wish to maximize diversity of object types.

I believe a linear equation expressing this would look like the following?

$a_{c}\cdot \left( \begin{array}{ccc} \\ 0 \\ 1 \\ 2 \\ .. \\ y \end{array} \right) + b_{c}\cdot \left( \begin{array}{ccc} \\ 0 \\ 1 \\ 2 \\ .. \\ y \end{array} \right) + .. +n_{c}\cdot \left( \begin{array}{ccc} \\ 0 \\ 1 \\ 2 \\ .. \\ y \end{array} \right) = T $

I've tried what little linalg I remember from university, plus some simple combinatorics.. no dice.