jmc

Question

How many ways are there to arrange 6 beads of distinct colors in a 2 3 grid if reflections and rotations are considered the same? (In other words, two arrangements are considered the same if I can rotate and/or reflect one arrangement to get the other.)

Accepted Answer

There are 6! = 720 ways to put the beads on the grid ignoring distinguishability. On the other hand, there are 4 possible transformations of the board using rotations and reflections (including the identity): arrayccccccc A & B & C & & C & B & A D & E & F & & F & E & D arrayarrayccccccc F & E & D & & D & E & F C & B & A & & A & B & C arrayNone of these transformations besides the identity fixes an arrangement, so each arrangement is equivalent to three others. As a result, there are 7204 = 180 different arrangements.