Answer:
3,360
Explanation:
so, to start with, there are 8 over 5 permutations (the sequence of the picked beads matters, but no repetitions of beads are possible) to pick 5 items out of 8 available ones.
that is
8! / ((8-5)!) = 8! / 3! = 8×7×6×5×4 = 6,720
rotations and reflections are the same thing here for a 1- dimensional sequence.
1 2 3 4 5 rotated around the middle (3) is
5 4 3 2 1
a reflection is again
5 4 3 2 1
so, we need to consider that instead of 8 beads for the first choice we have only 4 beads to choose from to leave the other 4 beads as choices for the last position.
once we have this established, it is sure that the first and the last position cannot have mirrored beads in any permutation. and therefore rotational or reflectional permutations are impossible.
in other words, half of the possible permutations would be rotations/reflections, and we need to eliminate them.
so, the calculation is
4×7×6×5×4 = 8×7×6×5×4/2 = 6,720/2 = 3,360