Question:
How many words can be formed with the letters of the word 'UNIVERSITY', the vowels remaining together?
Solution:
The word UNIVERSITY consists of 10 letters that include four vowels of which two are same.
Thus, the vowels can be arranged amongst themselves in $\frac{4 !}{2 !}$ ways.
Keeping the vowels as a single entity, we are left with 7 letters, which can be arranged in 7! ways.
By fundamental principle of counting, we get,
Number of words $=7 ! \times \frac{4 !}{2 !}=60480$