Originally Posted by

**amyw** I though about this problem a lot and came up wit the way to solve it, so I though I would share. Your help got me started, but wasn't completely right.

Because there are 12 bead, we think of the do-decagon and it group of symmetries. There are 24 in the symmetric group so that's a place to start. There are 12 rotations, the identity being one and 12 flips or reflections. There are six that flips from vertex to vertex and six that flip from mid-point to mid-point. So now I need the number fixed be each symmetry. Because there are 7 of one color and 5 of another the only rotation that fixes colors is the identity. (This would not be true if both number of beads were even.) That total is 12!/7!5!= 792. Of the flips, the mid point to midpoint flips fix nothing. (I had to draw this out to convince myself, but it's true.) The vertex to vertex flip, remember there are six, do fix things, but only when the colors are at opposite poles. There are 6 ways to do this. So on each of those place one red and one white at each vertices and then figure out the ways to place the other beads. I cut it in half and though of the five place on each side of the vertices and came up with (5!/3!2!)=60 then multiplied that by 2 for the other side to get 120. So the number of orbits equals (792+0+120+0)/24=38. So, there are 38 distinct necklaces that can be made from 7 beads of one color and 5 of another.

P.S. My professor said I did this exactly right.

Yeah!