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Joanna has seven beads that she wants to assemble into a bracelet.
Three of the beads have the same color, and the other four all have different colors.
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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When we consider all possible permutations of 7 different beds in line,
the number of all permutations is 7! = 5040.
When we identify circular permutations, their number become 6! = 720.
When we identify permutations under reflection, only half different distinguishable
permutations remains, i.e. 720/2 = 360.
When we identify permutations for the same color beds, the number of permutations
reduces in 3! = 6 times,
so the final number of distinguishable arrangements becomes 360:6 = 60.
ANSWER. Under given conditions/restrictions, there are 60 distinguishable bracelets.
Solved.