Answer:
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
×
6
×
5
×
4
×
3
×
2
×
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
!
×
3
!
×
2
!
=
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
!
×
1
!
×
1
!
×
1
!
×
1
!
×
1
!
×
1
!
which is the same as the
7
!
as before.)
