There are 5 vowels in the given word, 1 E, 1 U, 1 A, 1 I, and 1 O. Since these vowels are occurring together, so consider them as one letter, and when this letter is combined with the remaining 7 letters, then we have 8 letters in all, which can be arranged in (8!)/(2!) ways. Corresponding to the arrangements, the 5 vowels can be arranged in 5! ways 
Hence, required no. of ways = (8!)/(2!) * 5! = 20160 * 5! = 2419200

Arranging the word "PERMUTATION" with vowels together

To find the number of ways in which the word "PERMUTATION" can be arranged such that all the vowels come together, we can treat the cluster of vowels (EUAIO) as a single entity. This reduces the problem to arranging the letters "PRTMNTPN" and the cluster of vowels "EUAIO" together.

Step 1: Arranging the consonants
The word "PERMUTATION" has 4 consonants: P, R, T, and N. We can arrange these consonants among themselves in 4! (4 factorial) ways.

4! = 4 x 3 x 2 x 1 = 24

So, there are 24 ways to arrange the consonants.

Step 2: Treating the cluster of vowels as a single entity
As mentioned earlier, we can treat the cluster of vowels (EUAIO) as a single entity. Now, we have 5 entities to arrange: P, R, T, N, and the cluster of vowels.

5! = 5 x 4 x 3 x 2 x 1 = 120

So, there are 120 ways to arrange the 5 entities.

Step 3: Arranging the vowels within the cluster
Within the cluster of vowels, we have 5 vowels: E, U, A, I, and O. We can arrange these vowels among themselves in 5! (5 factorial) ways.

5! = 5 x 4 x 3 x 2 x 1 = 120

So, there are 120 ways to arrange the vowels within the cluster.

Step 4: Combining the arrangements
To find the total number of arrangements, we need to multiply the number of arrangements from step 1, step 2, and step 3.

Total number of arrangements = 24 x 120 x 120 = 345,600

Therefore, there are 345,600 ways to arrange the word "PERMUTATION" such that all the vowels come together.

Summary:
- Arrange the consonants (P, R, T, N) among themselves: 4!
- Treat the cluster of vowels (EUAIO) as a single entity: 5!
- Arrange the vowels (E, U, A, I, O) within the cluster: 5!
- Multiply the results from the above steps to get the total number of arrangements: 4! x 5! x 5! = 345,600 ways.