Question

How many permutations are there for the word numbers?

Answer 1

`7!`, which is 5040.

Explanation:

Since all letters are unique, we have 7 options for what letter goes first, then the remaining 6 options for what's 2nd, etc.

The total number of permutations is the product of how many options there are for each position:

`7! = 7 xx 6 xx 5 xx 4 xx 3 xx 2 xx 1 = 5040`

If some letters are duplicated, we would divide by the number of ways to permute those duplications, after permuting all the letters.

Example: How many ways can the letters of BANANA be permuted?

We have 1 B, 3 A's and 2 N's. So the number of permutations for BANANA are:

`(6!)/(1!xx3!xx2!)=60`

(Notice how the 6 on top equals the sum of the 1, 3, 2 on bottom. This is technically what we did for the letters in NUMBERS above, but since all letters were unique, we did

`(7!)/(1!xx1!xx1!xx1!xx1!xx1!xx1!)`

which is the same as the `7!` as before.)