SOLUTION: how many different words can be formed with the letter of the word "vicechancellor" so that the vowels are together?

"vicechancellor"

The vowels i,e,a,e,o can be arranged  ways.
We divide by 2! because of the 2 indistinguishable e's

For each of those ways of selecting the arrangement of the
vowels, it is the same as selecting the number of ways of
arranging the letters of

"vcchncllrX"

where X stands for the arrangement of the vowels which 
must be together:

That is .  We divide by 3! because of the 3
indistinguishable c's and we divide also by 2! because of the
2 indistinguishable l's.  So the desired number is

   

Therefore the answer is 18,144,000.

Edwin