Arrange the letters of the word ''DARKER'' so that the three vowels do not appear together. In how many ways can this be done?

'DARKER' has 6 letters.

Thus, we can arrange 6 letters in 6! ways.
But R gets repeated. There are 2R's. So divide by 2!

∴ Total ways = 6!/2! = 360
Vowels not together = Total ways - Vowels together

Consider the 2 vowels (A and E) as one group.
We have 4 letters and 1 group = 5
We can arrange them in 5! Ways.

But again here R comes twice. So we will have 5!/2!
Also, the 2 vowels can be arranged in 2! Ways.
SO the number of ways with vowels together = 2! × (5!/2!) = 120

∴ Number of ways with vowels not together = 360 - 120 = 240.