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Step 1: Calculate the total permutations of the letters in "VOWELS"
There are 6 distinct letters: V, O, W, E, L, S. The total number of ways to arrange 6 distinct letters is given by:
6! = 720 .
Step 2: Calculate the permutations where all 4 consonants are together
The consonants are V, W, L, and S. Treat them as a single block plus the 2 remaining vowels (O and E). Thus, we are effectively arranging 3 entities: (block of 4 consonants), O, and E.
The number of ways to arrange these 3 entities is 3! . Within the consonant block, the 4 consonants can be arranged among themselves in 4! ways. Therefore, the total number of arrangements with all consonants together is:
3! \times 4! = 6 \times 24 = 144 .
Step 3: Subtract the disallowed arrangements from the total
We do not want all consonants together. So, the required count is the difference between the total permutations and the permutations where all consonants are together:
6! - (3! \times 4!) = 720 - 144 = 576 .
Final Answer
The number of six-letter words formed using the letters of "VOWELS" such that all consonants do not come together is \boxed{576} .
