In how many different ways can the letters of the word 'CORRUPTION' be arranged so that the vowels always come together?

Question: In how many different ways can the letters of the word 'CORRUPTION' be arranged so that the vowels always come together?

Solution:
In the word 'CORRUPTION', we treat the vowels OUIO as one letter.

Thus, we have CRRPTN (OUIO).

This has 7 (6 + 1) letters of which R occurs 2 times and the rest are different.

Number of ways arranging these letters = 7!/2! = 2520.

Now, 4 vowels in which O occurs 2 times and the rest are different, can be arranged in 4!/2! = 12 ways.

Required number of ways = (2520 x 12) = 30240.