How many different bracelets have 3 identical orange beads, 3 identical black beads, and 1 teal bead, if rotating or flipping a bracelet does not change it?
Is the answer 720 or 140? I have no idea
I think it is only 10 ways.
Put the teal bead on first. It is the marker bead. This takes care of the rotation aspect.
Then you have 2 lots of 3 identical beads. so that is 6!/(3!3!) ways to place them. Which is 20 ways.
But the braclet can be flipped so I think this will halve the number of possibilities
For instance TOOOBBB is the same as TBBBOOO
So I think the answer is 10 ways.