Five beads of five different colours are threaded on a circular bracelet. Two bracelets count as the same if one can be turned or flipped into the other. How many different bracelets are there?
Hints
Take them one at a time. The first gives nothing away.
1A nudge
Count the arrangements, then count the repeats.
2The strategy
Start from all the arrangements as if the positions were fixed. Then work out how many of those are really the same bracelet: each one appears once per turn and again for each turn of the flipped version.
3The full solution
There are 120 arrangements. Each bracelet appears 5 times from turning and twice over from flipping, so 10 times in all: 120 divided by 10 = 12.
Solution
The reliable way
Overcount deliberately, then correct. Five distinct beads in five fixed positions give 5 x 4 x 3 x 2 x 1 = 120 arrangements. But turning the bracelet by one, two, three or four positions gives the same bracelet, and so does flipping it over and turning it. That is 5 turns times 2 for the flip = 10 arrangements per genuine bracelet. So 120 divided by 10 = 12. The transferable idea: name exactly how many times each object was counted before dividing.
The elegant way
Hold one bead fixed to kill the turning, leaving 4 x 3 x 2 x 1 = 24 arrangements of the rest, then halve for the flip: 12.
Why this is on the test: Deliberate overcounting followed by an exact correction is a genuine 5-point idea, and guessing the correction factor is what separates a right answer from a plausible one.