Number of Ways to Arrange String with Repeated Elements

Question

The letters A, A, B, B, C, C, D, D, are arranged in a straight line. How many arrangements are possible?

Also, the answer (from the answer key) is 2520. The problem is I don't have a clue how they got this?!

Answer 1

The number of ways to arrange $n$ items is $n!$. So the number of ways to arrange $8$ items would be $8!=40320$. However, you have $2!$ ways to arrange the two $A$'s. Since those two arrangements are identical, you need to divide your original answer by $2!$. Similarly you need to divide by $2!$ for the other pairs of letters giving you a final answer of $$\frac{8!}{2!\times2!\times2!\times2!}$$

Generally the number of ways to arrange a string of letters or numbers is the total number of items factorialed, divided by the factorial of the number of occurrences of each element. Note that $1!$ is trivial and doesn't need to be dealt with.

Finally, for another example, the number of ways to arrange the letters SENSES would be $$\frac{6!}{3!\times2!\times1!}$$