To find the number of permutations of the letters in the word "PERMUTATION" such that no two consecutive letters are both vowels or both identical, we will follow these steps:
### Step 1: Identify the letters in "PERMUTATION"
The word "PERMUTATION" consists of:
- Consonants: P, R, M, T, T, N (6 consonants)
- Vowels: E, U, A, I, O (5 vowels)
### Step 2: Calculate the arrangements of consonants
The consonants P, R, M, T, T, N can be arranged. Since the letter T repeats, the number of arrangements is given by:
\[
\text{Arrangements of consonants} = \frac{6!}{2!}
\]
Where \(6!\) is the factorial of the total consonants and \(2!\) accounts for the repetition of T.
### Step 3: Determine positions for vowels
Once the consonants are arranged, we can identify the positions for the vowels. Arranging the 6 consonants creates 7 potential slots for vowels (before the first consonant, between consonants, and after the last consonant).
### Step 4: Choose positions for the vowels
We need to choose 5 out of these 7 slots to place the vowels. The number of ways to choose 5 slots from 7 is given by:
\[
\binom{7}{5}
\]
### Step 5: Arrange the vowels
The 5 vowels can be arranged in the chosen slots in \(5!\) ways since all vowels are distinct.
### Step 6: Calculate total arrangements for the first case
The total arrangements for the first case (no two consecutive letters are both vowels) is:
\[
\text{Total arrangements} = \frac{6!}{2!} \times \binom{7}{5} \times 5!
\]
### Step 7: Calculate arrangements for the second case (identical letters)
In the second case, we treat the two T's as one unit. Thus, we have:
- Consonants: P, R, M, N, TT (5 consonants)
The arrangements of these consonants is:
\[
5!
\]
### Step 8: Determine positions for vowels in the second case
With 5 consonants arranged, we now have 6 slots for the vowels (similar to Step 3).
### Step 9: Choose positions for the vowels in the second case
We can still choose 5 out of these 6 slots for the vowels:
\[
\binom{6}{5}
\]
### Step 10: Arrange the vowels in the second case
The vowels can still be arranged in \(5!\) ways.
### Step 11: Calculate total arrangements for the second case
The total arrangements for the second case (where identical letters are consecutive) is:
\[
\text{Total arrangements (identical)} = 5! \times \binom{6}{5} \times 5!
\]
### Step 12: Calculate the required arrangements
To find the required number of arrangements such that no two consecutive letters are both vowels or both identical, we subtract the second case from the first case:
\[
\text{Required arrangements} = \left(\frac{6!}{2!} \times \binom{7}{5} \times 5!\right) - \left(5! \times \binom{6}{5} \times 5!\right)
\]
### Step 13: Simplify and calculate the final answer
Calculating the above expression will yield the final answer.
### Final Answer:
The final answer will be \(57 \times 5! \times 5!\).
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