How many words can be made by rearranging letters of VOWELS such that all consonants are not together

Answer

To solve the problem of how many words can be made by rearranging the letters of "VOWELS" such that all consonants are not together, we can follow these steps:

### Step 1: Identify the letters
The word "VOWELS" consists of the following letters:
- Vowels: O, E
- Consonants: W, L, S

### Step 2: Calculate the total arrangements without restrictions
The total number of letters in "VOWELS" is 6. The total arrangements of these letters can be calculated using the factorial of the number of letters:

\[ \text{Total arrangements} = 6! = 720 \]

### Step 3: Calculate arrangements with consonants together
To find the arrangements where all consonants (W, L, S) are together, we can treat the group of consonants as a single entity or letter. Thus, we have the following "letters":
- Group of consonants: (WLS)
- Vowels: O, E

This gives us a total of 4 "letters" to arrange: (WLS), O, E.

The number of arrangements of these 4 "letters" is:

\[ \text{Arrangements with consonants together} = 4! = 24 \]

### Step 4: Arrange the consonants within their group
The consonants W, L, S can be arranged among themselves in:

\[ \text{Arrangements of consonants} = 3! = 6 \]

### Step 5: Calculate total arrangements with consonants together
Now, we multiply the arrangements of the 4 "letters" by the arrangements of the consonants:

\[ \text{Total arrangements with consonants together} = 4! \times 3! = 24 \times 6 = 144 \]

### Step 6: Calculate arrangements with consonants not together
To find the arrangements where the consonants are not together, we subtract the arrangements where they are together from the total arrangements:

\[ \text{Arrangements with consonants not together} = \text{Total arrangements} - \text{Arrangements with consonants together} \]

Substituting the values we calculated:

\[ \text{Arrangements with consonants not together} = 720 - 144 = 576 \]

### Final Answer
Thus, the number of words that can be formed by rearranging the letters of "VOWELS" such that all consonants are not together is:

\[ \boxed{576} \]