How many ways can you arrange the letters of the word full?

Answer 1

To mathematically solve this:

Four letter word: n = 4

One F: L1 = 1

One U: L2 = 1

Two L's: L3 = 2

For the next step, you need to know what a factorial is. A factorial (n! - you'd say "n factorial") can be simply described as the product of that integer by all integers lower than it down to 1.

Example

5! = 5 x 4 x 3 x 2 x 1 (start with 5 and go all the way down to the optional 1)

We're going to be dividing the length factorial by each letter factorial:

n! / ( L1 x L2 x L3 )

4! / ( 1! 1! 2! ) = ( 4 x 3 x 2 x 1 ) / ( 1 x 1 x 2 x 1 ) = 4 x 3 = 12

With such a short word, it's easy enough to list the combinations. I'll do this so you can verify the answer. Breaking into cases always makes this easier.

Start with F:

FULL

FLUL

FLLU

Start with U:

UFLL

ULFL

ULLF

Start with L:

LUFL

LFUL

LLUF

LLFU

LULF

LFLU

There are 12 combinations, because you treat both L's as the same character.