Okay, so I know that if 15 beads, 5 red, 5 yellow, 5 blue, the number of possible combinations is: 14! / 2*5!*5!*5!
But say all 15 were of the same colour. (the answer is obviously 1). But wouldn’t that be 14! / 2 * 15! ? Because of the (n-1)! For a circle, divide by 2 for flipping, then the 15! for beads of the same colour. I know it’s super trivial, but I don’t understand what’s wrong with it. Thanks
The actual calculation would be 15!/15! = 1 (which you know to be the correct final result).
Where your error comes from is by just applying formulaic rules without thinking through what those are saying, specifically about distinguishable and indistinguishable objects. So lets expand on this.
To start: 15 beads can be arranged in 15! permutations.
Because this is a circle, when you have multiple colors you can create 15 distinguishable permutations that differ only by rotation. This gives you 15!/15 permutations which is the same as 14! or (n-1)! But with only one color, you don't have distinguishable rotations. You also can flip the circle, with different colors then having distinguishable permutations differing only by flipping, thus divide by 2. With only one color, the flip is indistinguishable, so no division.
The key is "distinguishable".
Finally is within each distinct color set, the ways those indistinguishable beads can be arranged is (color quantity)!
Thus with only one color, you have 15! permutations (no distinguishable rotations or flipping), divided by 15! permutations of that color, e.g. 15!/15!