Question

How many different arrangements of 5 letters can be made using the letters in the word "FLOOR"?

Answer 1

There exist 2 cases to consider :

Case 1 - The two O's are not identical

Then it is equivalent to asking in how many different ways can 5 different units be arranged in a row.
This is simply `5! = 120` different ways.

Case 2 - The two O's are identical

This is then equivalent to asking in how many different ways can 5 items be arranged in a row, if 2 of the 5 items are identical.

By the addition principle, there are 10 possible ways to place the 2 identical objects in the 5 available slots in the row.

Now for each of these 10 different arrangements of the identical items, the other 3 non-identical items may be placed and arranged in `3!=6` different ways.

So in total, there are then `6xx10=60` different ways.

Note that since the 2 identical objects can themselves be arranged in `2!` different ways, we don't include this in the total since these arrangements are considered identical to one of the others already.