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Joanna has several beads that she wants to assemble into a bracelet. There are seven beads: five of the beads have the same color,
and the other two all have different colors. Using all seven beads, in how many different ways can Joanna assemble her bracelet?
(Two bracelets are considered identical if one can be rotated and/or reflected to obtain the other.)
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As I understand from the context, five beads of the same color A are indistinguishable,
while two remaining beads have different colors B and C, what are different from A, too.
The question is about the number of distinguishable permutations/arrangements.
Using rotations, we can bring the unique bead of the color B to position "12 o'clock".
Then the other 6 breads occupy the remaining 6 positions on the circle.
The bead of color C can occupy any of remaining 6 positions, and it makes the only difference
in circular placing the beads. So, there are only 6 different (distinguishable) circular permutations.
If to consider reflected configurations as identical, then 6 different circular permutations
produce only 3 different distinguishable arrangements.
ANSWER. There are 3 different circular arrangements, if to consider reflected arrangements as identical.