To solve the problem of finding the number of permutations of the letters in the word "PERMUTATION" such that no two consecutive letters are both vowels or both identical, we can follow these steps:
### Step 1: Identify the letters in the word
The word "PERMUTATION" consists of:
- Vowels: E, U, A, I, O (5 vowels)
- Consonants: P, R, M, T, T, N (6 consonants)
### Step 2: Calculate the arrangements of consonants
The consonants are P, R, M, T, T, N. Since T is repeated, the number of arrangements of the consonants is given by:
\[
\text{Arrangements of consonants} = \frac{6!}{2!} = \frac{720}{2} = 360
\]
### Step 3: Determine positions for vowels
When the consonants are arranged, they create gaps for the vowels. For 6 consonants, there are 7 possible positions (gaps) to place the vowels:
- _ C _ C _ C _ C _ C _ C _ C _ (where C represents a consonant)
### Step 4: Choose positions for the vowels
We need to choose 5 positions out of the 7 available for the vowels. This can be done in:
\[
\text{Ways to choose positions} = \binom{7}{5} = \binom{7}{2} = 21
\]
### Step 5: Arrange the vowels
The vowels E, U, A, I, O can be arranged in the chosen positions. The number of arrangements of the 5 distinct vowels is:
\[
5! = 120
\]
### Step 6: Calculate total arrangements without restrictions
The total arrangements without considering the restrictions on vowels is:
\[
\text{Total arrangements} = \text{Arrangements of consonants} \times \text{Ways to choose positions} \times \text{Arrangements of vowels}
\]
\[
= 360 \times 21 \times 120
\]
### Step 7: Calculate arrangements where identical letters are together
Now, we need to consider the case where the identical letters (the two T's) are treated as one unit. This reduces the consonants to P, R, M, TT, N (5 units). The arrangements of these consonants are:
\[
\text{Arrangements of consonants (with TT)} = 5! = 120
\]
The number of gaps for vowels remains the same (6 gaps). The number of ways to choose 5 positions from 6 is:
\[
\binom{6}{5} = 6
\]
And the arrangements of the vowels remains:
\[
5! = 120
\]
So, the total arrangements where the identical letters are together is:
\[
\text{Total arrangements (identical together)} = 120 \times 6 \times 120
\]
### Step 8: Subtract the identical arrangements from total arrangements
Finally, we subtract the arrangements where identical letters are together from the total arrangements:
\[
\text{Required arrangements} = \text{Total arrangements} - \text{Total arrangements (identical together)}
\]
### Final Calculation
1. Calculate total arrangements:
\[
360 \times 21 \times 120 = 907200
\]
2. Calculate arrangements with identical letters together:
\[
120 \times 6 \times 120 = 86400
\]
3. Calculate the required arrangements:
\[
907200 - 86400 = 820800
\]
Thus, the number of permutations of the letters of the word "PERMUTATION" such that no two consecutive letters are both vowels or both identical is **820800**.
