Pick's theorem application In a game let us say that we have $3$ ways to score points, getting $1,2 \text{ or } 3$ points. I make a total of $30$ points. What are the various ways I can make $30$ points? 
If we plug in values, we have $a + 2b + 3c = 30$ and we have to find ordered pairs $a,b,c$ where $a,b,c$ are whole numbers. However i saw another way of solving this question, using Pick's theorem. I found it better because plugging in would be difficult if it would have been $300$ in place of $30$.
According to the solution I read, the number of $2$s and $3$s will affect the number of $1$s, so we can make represent the $2$s on the x-axis and the $3$s on the y-axis and vice-versa. So how can I use the Pick's theorem to solve this question?

I don't know why you would want to use Pick's theorem to solve this, but you are trying to find the number of lattice points satisfying your equation. These lie in a triangle, so if you project it into the $x,y$ plane the number of lattice points stays the same, and you can use use pick's theorem (assuming you can compute the number of boundary points -- this is easier, since dimension is reduced by one. 
A completely mechanical way to answer these questions is given by Euler's generating function -- the number of solutions is the coefficient of $x^{30}$ in 
$$\frac{1}{1-x} \frac{1}{1-x^2} \frac1{1 - x^3}.$$ You can write down the general term in closed form using partial fractions, but I leave this to you...
