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

810

1440

2880

50400

Explanation:

In the word CORPORATION, we treat the vowels OOAIO as one letter.

Thus, we have CRPRTN [OOAIO].

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

Number of ways arranging these letters =$ \dfrac{7!}{2!} $= 2520.

Now, 5 vowels in which O occurs 3 times and the rest are different, can be arranged

in$ \dfrac{5!}{3!} $= 20 ways.

$\therefore$ Required number of ways = $\left(2520 \times 20\right)$ = 50400.