The number of six-letter words, with or without meaning, formed using all the letters of the word 'VOWELS' so that none of the consonants come together is 144.
To find the number of six-letter words formed using all the letters of the word 'VOWELS' such that all the consonants never come together, we can use the concept of permutations.
The word 'VOWELS' has 6 letters, with 2 vowels (O and E) and 4 consonants (V, W, L, and S). Out of the 6 positions in the word, there are 2 positions where vowels can be placed. Since the consonants cannot come together, we need to find the number of arrangements for the consonants and multiply it by the number of arrangements for the vowels.
Let's start with the consonants. There are 4 consonants, and we need to place them in 4 positions such that they don't come together. This can be done using the concept of permutations with restrictions. We can consider the vowels (O and E) as a single entity and find the number of arrangements for this entity and the consonants together.
So, the number of arrangements for the consonants and vowels together is 4! * 3! = 144.
Therefore, the number of six-letter words formed using all the letters of the word 'VOWELS' such that all the consonants never come together is 144.