Removing one coin from each of the triangles — more specifically, one a, one b, and one c — forms a pattern (like the example shown).
Every possibility is listed below (ignore the circles and squares around some of them).
How can you systematically list all of the possibilities, so that you get that answer?
We have to derive the underlying structure of the problem, forgetting about the artistic paraphernalia. We have three lines of $a$,$b$, $c$ and should select one element from each numbered line and one element from each letterized column: $$\left[\matrix{a_1&b_1&c_1\cr a_2&b_2&c_2\cr a_3&b_3&c_3\cr}\right]\ .$$ As in a $3\times3$-determinant there are $6$ ways to do this: Select one object from the first line ($3$ ways), then one allowed element from the second line ($2$ ways) and then the only remaining element from the third line.
You can arrange the admissible selection lexicographically as follows: $$a_1b_2c_3,\ a_1b_3c_2, \ b_1a_2c_3, \ b_1c_2a_3, \ c_1a_2b_3, \ c_1b_3a_2\ .$$