How do we deal with Permutations with Repeats?

Example

In how many distinct ways can the letters of the word EGG be arranged?

Expectation: Simply do the following $3 \times 2 \times 1 = 6$.

But this doesn't work. There are only THREE distinct arrangements: EGG, GEG, and GGE. Why? Because of that damn repeated G.

The Formula

$$\frac{n!}{(repeat_1)!(repeat_2)!(repeat_3)!}$$

This formula accounts for repeats. In our original example, notice that the G was repeated twice, so the formula becomes:

$$\frac{3!}{2!} = 3$$

Example 2

How about in the case of GOOGLE? Notice that there are two Gs and two Os, so the formula is:

$$\frac{6!}{2!2!} = 180$$

Example 3

What about something silly like AAABBBCC? Notice that there are three As, three Bs, and two Cs, so the formula becomes:

$$\frac{8!}{3!3!2!}=560$$