How many ways are there to arrange the letters in ALABAMA?

Answer 1

There are 210. Although the number of possible combinations of any 7 elements is (7!) or 7 factorial - 7 x 6 x 5 x 4 x 3 x 2 x 1 or 5040, there is no way to distinguish the identical elements, which are the 4 A's. So the number of combinations in this spelling exercise is (7!) divided by (4!), 5040/24 = 210.

As an equation: (7!)/(4!) = (7 x 6 x 5) = 210

By Arrangement

If you looked at the A's as separators, and then calculated the number of ways that the 3 remaining letters could be fit into the 5 spaces, it would look something like this:

_A_A_A_A_

LBMA_A_A_A_

ALBMA_A_A_

A_ALBMA_A_

and so forth.

Grouped together, L B and M can be arranged in 6 distinct ways. There are

5 ways to keep the three letters together along with the separators, so that

makes 30 possibilities.

There are 20 ways to have 2 letters of the three together, with the third

someplace else. (5 different locations for a group of two, leaving four possible

locations for the remaining letter) In each of the 20 configurations the 3 letters

can still be arranged in 6 distinct ways, making 120 more possibilities.

So there are 10 different ways to keep the three letters separate from each other, and each of the 10 ways can still have the three letters in 6 different arrangements, giving 60 more possibilities.

30 + 120 + 60 = 210.