Final answer:
The number of six-letter words formed from the word 'VOWELS', with all consonants never together, is calculated by subtracting the number of arrangements with consonants together from the total arrangements, giving a result of 576 possible words.
Step-by-step explanation:
The question pertains to the calculation of a combinatorial problem involving the formation of six-letter words from the word 'VOWELS', under the restriction that all consonants do not appear together. The word 'VOWELS' contains 4 consonants (V, W, L, S) and 2 vowels (O, E). To ensure that all consonants do not appear together, we can employ a combinatorial approach to first determine the total number of arrangements without restriction and then subtract the number of arrangements where all consonants are together.
Without restrictions, each of the 6 different letters in the word 'VOWELS' can be arranged in 6! (factorial) ways, which equals 720 arrangements. To find the number of arrangements where all consonants are together, we can treat the consonants (VWLS) as one single unit along with the vowels (O and E). So, we are arranging three units (VWLS, O, E), which can be done in 3! ways. Additionally, the block of consonants (VWLS) can be arranged amongst themselves in 4! ways. Thus, the number of invalid cases (where consonants are together) is 3! * 4! = 6 * 24 = 144. Finally, to find the number of valid arrangements where the consonants are never together, we subtract the invalid cases from the total cases: 720 - 144 = 576.
The letters crossed off as always or sometimes vowels are: <a>, <e>, <i>, <o>, <u>, <y> and <w>. The remaining ones are always consonants. Therefore, the possible number of six-letter words formed from the word 'VOWELS', such that all the consonants never come together, is 576.