# Problem of the Week Problem D The Dart Game

A carnival dart game has three non-overlapping circles in a rectangle. One circle has a value of $$2$$, another has a value of $$3$$, and the third has a value of $$5$$. You are allowed to throw up to $$10$$ darts, and you start the game with a running total of 0. If a dart lands in one of the circles, you add the value of the circle to the running total. If a dart does not land in one of the circles, then you do not add anything to the running total for that throw.

Suppose you have exactly $$30$$ points after $$10$$ throws. Let $$a$$ represent the number of throws that landed in the circle with value $$5$$, let $$b$$ represent the number of throws that landed in the circle with value $$3$$, and let $$c$$ represent the number of throws that landed in the circle with value $$2$$. Determine all possibilities for $$(a,b,c)$$.