Question

In how many ways can the letters in the word ALLERGENS be arranged?

Answer 1

`(9!)/((2!)(2!))=362880/4=90,720`

Explanation:

If we had a word with 9 different letters (or simply asked the question "how many numbers can be made rearranging the numbers 1 through 9"), we'd have a straightforward factoral problem, where the solution would be `9! =362,880`.

In this case, however, we have duplicate letters - 2 e and 2 l - and these are interchangeable, and so we need to eliminate the duplicate counting from these 2 letters.

For each letter that has duplicates, we divide by the factoral of the number of each duplicate letter - and so we'll divide by `2!` for the 2 letter e and divide by `2!` again for the 2 letter l. That gives us:

`(9!)/((2!)(2!))=362880/4=90,720`