Question

In the figure below, $3$ of the $6$ disks are to be painted blue, $2$ are to be painted red, and $1$ is to be painted green. Two paintings that can be obtained from one another by a rotation or a reflection of the entire figure are considered the same. How many different paintings are possible?

$6$

$8$

$9$

$12$

$15$

Solution:

The figure has $3$ corner disks and $3$ non-corner disks, with the symmetry group of a triangle. Fix the green disk's type. If green is a corner, the two red disks can be arranged so that both, one, or neither is adjacent to green, giving $1 + 3 + 2 = 6$ distinct paintings. If green is a non-corner, the two reds can have both, one, or neither in a corner, again $1 + 3 + 2 = 6$ paintings. The blue disks fill the rest, so the total is $6 + 6 = 12.$

Thus, the correct answer is D.

2017 AMC 12B Problem 13

Solution:

Problem 13 in Other Years