In how many ways can the letters of the word ' PERMUTATIONS' be arranged if the
vowels are all together

Answer

To solve the problem of arranging the letters of the word "PERMUTATIONS" with the condition that all the vowels are together, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the letters in the word**:
The word "PERMUTATIONS" consists of 12 letters. The letters are:
P, E, R, M, U, T, A, T, I, O, N, S.

2. **Identify the vowels and consonants**:
The vowels in the word are: E, U, A, I, O (total of 5 vowels).
The consonants are: P, R, M, T, T, N, S (total of 7 consonants).

3. **Group the vowels together**:
Since we want all the vowels to be together, we can treat the group of vowels (E, U, A, I, O) as a single unit or letter. This gives us:
- Vowel group: (EUAIO)
- Remaining consonants: P, R, M, T, T, N, S.

Now we have a total of 8 units to arrange:
- 1 vowel group + 7 consonants = 8 units.

4. **Calculate the arrangements of the 8 units**:
The total arrangements of these 8 units (considering that T is repeated) is given by:
\[ \frac{8!}{2!} \]
where \(8!\) is the factorial of the total units and \(2!\) accounts for the repetition of the letter T.

5. **Calculate the arrangements of the vowels within their group**:
The vowels E, U, A, I, O can be arranged among themselves in:
\[ 5! \]
since all the vowels are distinct.

6. **Combine the arrangements**:
The total number of arrangements of the letters in "PERMUTATIONS" with the vowels together is:
\[ \text{Total arrangements} = \frac{8!}{2!} \times 5! \]

7. **Calculate the values**:
- \(8! = 40320\)
- \(2! = 2\)
- \(5! = 120\)

Therefore,
\[ \text{Total arrangements} = \frac{40320}{2} \times 120 = 20160 \times 120 = 2419200 \]

### Final Answer:
The total number of ways to arrange the letters of the word "PERMUTATIONS" with all the vowels together is **2419200**.