Question 808512: I have a question that says: How many 4 letter arrangements can be made from the word ALASKA. The answer says 72, but I have no clue how to solve it..
Answer by KMST(5328)   (Show Source):
You can put this solution on YOUR website! There's probably a better way to do it, and the kids at artofproblemsolving.com probably learn it. For this kind of problem, asking that problem in the middle-schoolers forum would get you an answer, faster than asking a roomful of engineers.
Here's my way to get to the answer anyway.
Any 4-letter arrangement would use at least one A, but you can also use two A's, or all three A's.
Using just one A, you need to use all of the the 3 remaining letters (L, S, and K).
There are 6 different orders in which those letters may appear in the 4-letter arrangement (permutations of 3).
For each of those 6 options, there are 4 places to stick the A:
as the first letter, as the second one, as the third, or as the fourth.
The choices multiply to give you  4-letter sequences having one A.
Using two A's, you need to use two of the the 3 remaining letters (L, S, and K).
Considering which two letters to use and in what order they will appear, your choices are
 .
The two A's could appear in different places in the 4-letter arrangement.
As before A could have 4 places to appear, and you have to choose 2 of those places (combinations of 4 choosing 2).
That gives you
 choices.
Again, the possibilities for the A's, and the possibilities for the other letters multiply to give you
 4-letter arrangements having two A's.
Finally, if you use all 3 A's, you can only choose
which of the other 3 letters to include, and
which of the 4 possible places to put that letter.
That gives you  4-letter arrangements having three A's.
Counting them all, you have  4-letter arrangements made from the letters in the word ALASKA.
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