2017 AMC 10B Problem 18

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

Answer: D

Concepts:combinations symmetry casework

Difficulty rating: 2010

Small hint:

Use symmetry to reduce the green disk to two cases

Big hint:

For each green position, count possible red-disk placements up to symmetry

Solution:

By symmetry, the green disk has two possible types of position: a corner or a side midpoint. Fix one representative of either type. There are

\binom52=10

ways to choose the two red disks.

The reflection that fixes the green position fixes one of the other disks and exchanges the other four disks in two pairs. Exactly

2

red-disk choices are unchanged by this reflection: choosing either exchanged pair. The other

8

choices form

4

mirror-image pairs. Hence there are

2+4=6

paintings for each type of green position.

The two types therefore give

6+6=12

paintings.

Thus, the correct answer is D .