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 Question from Anna, a student: How many arrangements of FATHER can be made if all the vowels must be kept together?

Hi Anna,

I will illustrate how to solve your problem using a different word.

How many arrangements of ALIVE can be made if all of the vowels must be kept together?

Notice that ALIVE has 5 letters, 3 of which are vowels. Since the vowels must be kept together, consider them as 1 object, which, together with the L and the V gives us 3 objects to arrange.

There are 3! = 6 ways to arrange these three objects. But the three vowels themselves, although kept together, could also be arranged in 3! = 6 ways themselves.

Therefore there are 3! x 3! = 6 x 6 = 36 arrangements possible.