$$x\_1 + x\_2 + x\_3 + x\_4 = 10$$
But $x\_1 \ge 1$ and $x\_2 \ge2$.

Finally, $x\_1<6, x\_2<6, x\_3<6$ and $x\_4<6$.

I started off by finding the total amount that you can get for the original equation. That would be $C (13, 3)$. Then i calculated for the cases of numbers being greater than or equal to 5. I dont know how to incorporate $x\_1\ge1$ and $x\_2\ge2$. Any help?

$x\_1, x\_2, x\_3, x\_4$ are all integers that could be >= 0(But $x\_1$ and $x\_2$ have their own restriction).