The number of words, which can be formed using all the letters of the word "DAUGHTER", so that all the vowels never come together, is :

Solution

Maths sir · 3 steps

Count all arrangements, then subtract the ones where A, U, E sit together as one block.

1. All arrangements. DAUGHTER has 8 letters, all different. They can be arranged in 8! = 40320 ways.
2. Vowels together. The vowels are A, U and E. Tie them into one block. That block and the 5 consonants D, G, H, T, R make 6 units, arranged in 6! = 720 ways. Inside the block the 3 vowels can be ordered in 3! = 6 ways, so there are 720 × 6 = 4320 such words.

   8! - 6! × 3! = 40320 - 4320 = 36000

3. So. There are 36000 words in which the three vowels are not all together, which is option (d).

The idea to keep. For "never all together", do not count it directly. Count the total, count the "all together" cases by tying them into one block (and arranging inside it), then subtract.

Answer from the NTA final answer key. Solution by PrepWiser.