Count Arrangements With Repeated Items

Count arrangements with repeated items

When some items are identical, not every rearrangement is new. Swapping two matching items does not create a different arrangement, so you must divide by the repeats.

1) Count the total positions

If there are (n) items altogether, start with the number of arrangements of (n) distinct items: (n!).

2) Correct for repeated items

If one item is repeated (r_1) times, another (r_2) times, and so on, divide by each repeated factorial:

[ \frac{n!}{r_1!,r_2!\cdots} ]

This gives the number of distinct arrangements.

3) Compute and simplify

Evaluate the factorials carefully, then simplify the fraction. If the answer is exact, leave it as a whole number.

4) Check your result

A quick check is to ask: if all items were different, would the answer be larger? It should be. Also, repeated items should reduce the count compared with (n!).

Common mistake

Do not divide by the number of repeated items themselves; divide by the factorial of each repeat count.

Use this same method every time: total factorial first, then divide for each group of identical items.