The number of six letter words (with or without meaning), formed using all the letters of the word 'VOWELS', so that all the consonants never come together, is ___________.

Solution

Maths sir · 4 steps

Count the words where all four consonants do come together, then take those away from all 720.

1. Sort the letters. The letters V, O, W, E, L, S are all different. The vowels are O and E, and the consonants are V, W, L and S, so there are 4 consonants.
2. All arrangements. Six different letters can be arranged in 6! = 720 ways.
3. Consonants together. Tie V, W, L, S into one block. The block, O and E make 3 units, which arrange in 3! = 6 ways. Inside the block the 4 consonants arrange in 4! = 24 ways, so there are 6 × 24 = 144 such words.

   6! - 3! × 4! = 720 - 144 = 576

4. So. The number of words in which the consonants are never all together is 576.

The common slip: 144. Stopping after counting the words with the consonants together gives 144. That is the case to remove, so subtract it from 6! = 720.

The idea to keep. When a question says a group must never all come together, count the arrangements where it is together and subtract them from the total. Do not forget to arrange the letters inside the block.

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