Question

How many different arrangements of the letters in the word BETTER are there?
180
360
720

Answer 1

would it be 720 because 6! = 720?

Answer 2

Well, that's including any duplicates since we have two T's. The number of different arrangements is going to be half that.

Answer 3

360

Answer 4

In general, you just divide off by the number of each letter in the word factorial.
Like, for a word like "diploid," we'd have 7 letters (7! arrangements) and divide off 2! for the d's and 2! for the i's. \( \large \frac{7!}{2! \times 2!} \)
In a word like "glasses," we'd again have 7 letters (7! arrangements) and divide off 3! for the s's. \( \large \frac{7!}{3!} \).

This is because if we have repeated letters, they can take each others' places and thus have n! arrangements for a letter repeated n times.
Since \(n!\) are duplicates and we only want one copy, we have to divide the total number of arrangements by the total duplicates to make the duplicates essentially cancel out in the final result.