Is this a Permutation Question or a Combination Question?
Today I had my middle schoolers working on some of the new MATHCOUNTS Club problems. One of them was the following:
A bracelet is made by stringing together four beads. Each bead is either red or green. How many different color patterns are possible for the bracelet, where patterns are considered the same if turning one will produce the other, as shown here?
Since we had just recently been working on Permutations and Combinations (and one day I may even get around to finishing the blog post I’ve been planning on that topic), one boy asked, “Is this a permutation question or a combination question?”
It was a question that had never occurred to me before, and I thought it was a great question. After thinking about that for a moment, I replied that it was sort of a permutation question, since the order of the beads does matter, but that since the permutations were arranged on circles, so that there was no fixed start or end of the sequence, it wouldn’t yield to a basic permutation approach (or for that matter a basic combination approach).
I suggested that in this case, “Make and Organized List” was probably the most fruitful strategy. But for a larger bracelet, that would break down, of course. Is there a good way to solve this problem for, say, a bracelet with 10 beads, each of which may be red or blue? It seems like you would need to still consider casework for the number of red (or blue) beads, and then figure permutations, adjusting for overcounting due to both identical items and rotational symmetry.