Bracelet Problem
Probability
Level
3
If you have 8 distinct beads, how many different bracelets can you make consisting of 8 beads, with one of each kind?
The answer is 2520.
This section requires Javascript.
You are seeing this because something didn't load right. We suggest you, (a) try
refreshing the page, (b) enabling javascript if it is disabled on your browser and,
finally, (c)
loading the
non-javascript version of this page
. We're sorry about the hassle.
1 solution
The first bead has 8 choices, the second bead has 7 choices, the third has 6, and so on. This is because once the first bead's color is chosen, the next bead has its options reduced by one, since all 8 beads are distinct. We are left then with 8! or 8x7x6x5x4x3x2x1 = 40320.
The problem is we have to account for duplicates and reflections.
Given a solution of colors, we could shift a bead to the other side of the bracelet and get the same sequence of colors but with a different initial color. For every solution, there are 7 duplicates, therefore we must divide by 8. 40320 / 8 = 5040
We must also take into account the reflections. We could simply take the bracelet off and put it back after flipping it and get a mirror image of the initial sequence. Therefore, we must also divide the result by 2. 5040 / 2 = 2520