Butcher Shop Prices

We just have to count the number of couples $(f,h)\in[0,100]^2$ such that $3f+2h \in [0,100]$ , that is: $S=\sum_{f=0}^{33}\left(\left\lfloor\frac{100-3f}{2}\right\rfloor+1\right).$ By splitting the sum over even and odd values of $f$ , we get: $S = 2\cdot\sum_{j=0}^{16}(50-3j) = 2\cdot\left(50\cdot 17-3\sum_{j=0}^{16}j\right) = 2\cdot\left(50\cdot 17 - 3\cdot8\cdot 17\right) = 2\cdot 17\cdot(50-24) = 884.$

We can determine the price of any combination $x(2t+f)+y(3t+h)$ where x and y are integers.

The boundary conditions are $0\leq x$ , $0 \leq y$ and $2x+3y \leq100$ .

So we need to count the grid points that lie within (or on the boundary of) the triangle with vertices $(0,0), (50,0), (0,33\frac{1}{3})$ .

Starting at the top row (y=33), subsequent rows have $1,3,4,6,7,9,...49,51$ gridpoints.

These numbers add up pairwise as $4+10+16+...+100$ and we can take the sum of these as $\frac{104×17}{2}=\boxed{884}$