Question 1191564: How many ways can you arrange the letters of the word EDUCATION, such that the vowels are always together?
Answer by math_tutor2020(3816)   (Show Source):
You can put this solution on YOUR website!
The vowels in the order presented are: E, U, A, I, O
or perhaps better sorted as: A, E, I, O, U
Think of the vowels as one block, and have some other letter (say X) representing that block.
We go from the word EDUCATION to X,D,C,T,N
The initial 9 letter word drops to 9-5 = 4 letters after we kick out the vowels.
Then introducing letter X brings the count up to 5 letters.
Anywhere you see an X, replace it with some permutation of A,E,I,O,U.
There are 5! = 5*4*3*2*1 = 120 ways to arrange X,D,C,T,N
And there are 5! = 120 ways to arrange the five vowels.
Overall, there are (5!)*(5!) = 120*120 = 14400 different arrangements such that the vowels stick together.
Answer: 14400
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