How many words can be formed by arranging the letters of the word 'ARRANGEMENT', so that the vowels remian together?

Answer

60480

Explanation

Let us assume that the vowels AAEE form a single letter.
Now, AAEE + RRNGMNT gives us 8 letters, out of which there are 2 R's, 2 N's, 1 G, 1 M and 1 T.
Number of these arrangements `=(8!)/((2!)xx(2!))=10080.`
Now, AAEE has 4 letters, out of which there are 2 A's and 2 E's.
Number of their arrangements `=(4!)/((2!)xx(2!))=6.`
Required number of words formed `=(10080 xx6)=60480.`