Permutations of a Multiset

This formula counts distinct ways to arrange a collection of items when some items are indistinguishable from each other. Start with n! ways to arrange all items as if they were different, then divide by each n_i! because swapping identical items within the same group does not create a new arrangement. In other words, it corrects the overcounting caused by repeated objects.

Use it when counting permutations of letters in a word with repeated letters, arranging people or objects where some are identical, or finding the number of distinct orderings in multistep arrangements with repeats. It is especially useful in counting problems where only the pattern matters, not which identical copy is which.

• Forgetting to divide by every repeated group, not just one of them
• Using the formula when all items are distinct, which reduces to n! instead
• Treating identical items as different and counting duplicate arrangements multiple times