To solve the problem of finding the total number of words that can be formed from the letters of the word "PERMUTATION" such that no vowel occupies any space between two consonants, we can follow these steps:
### Step 1: Identify the vowels and consonants in the word "PERMUTATION"
The word "PERMUTATION" consists of the following letters:
- Vowels: E, U, A, I, O (total 5 vowels)
- Consonants: P, R, M, T, T, N (total 6 consonants)
### Step 2: Arrange the consonants
Since we want to ensure that no vowel is placed between two consonants, we can start by arranging the consonants. The consonants are P, R, M, T, T, N.
To find the number of ways to arrange these consonants, we use the formula for permutations of multiset:
\[
\text{Number of arrangements of consonants} = \frac{n!}{p_1! \times p_2! \times \ldots \times p_k!}
\]
where \(n\) is the total number of items, and \(p_1, p_2, \ldots, p_k\) are the counts of indistinguishable items.
For our consonants:
- Total consonants = 6 (P, R, M, T, T, N)
- The letter T appears 2 times.
Thus, the number of arrangements of the consonants is:
\[
\text{Arrangements} = \frac{6!}{2!} = \frac{720}{2} = 360
\]
### Step 3: Determine the positions for the vowels
Once the consonants are arranged, we will have gaps where the vowels can be placed. For example, if we arrange the consonants as follows:
C1 C2 C3 C4 C5 C6
We can place the vowels in the gaps:
- _ C1 _ C2 _ C3 _ C4 _ C5 _ C6 _
This gives us 7 possible positions (gaps) to place the vowels.
### Step 4: Arrange the vowels
We have 5 vowels (E, U, A, I, O) to place in the 7 gaps. We need to choose 5 gaps out of the 7 available. The number of ways to choose 5 gaps from 7 is given by the combination formula:
\[
\text{Ways to choose gaps} = \binom{7}{5} = \binom{7}{2} = 21
\]
Now, we can arrange the 5 vowels in the chosen gaps. Since all vowels are distinct, the number of arrangements of the vowels is:
\[
\text{Arrangements of vowels} = 5! = 120
\]
### Step 5: Calculate the total arrangements
Finally, we multiply the number of arrangements of consonants, the number of ways to choose gaps, and the arrangements of vowels:
\[
\text{Total arrangements} = (\text{Arrangements of consonants}) \times (\text{Ways to choose gaps}) \times (\text{Arrangements of vowels})
\]
\[
\text{Total arrangements} = 360 \times 21 \times 120
\]
Calculating this gives:
\[
\text{Total arrangements} = 360 \times 21 = 7560
\]
\[
7560 \times 120 = 907200
\]
### Final Answer
Thus, the total number of words that can be formed is **907200**.
