To solve the problem of how many different words can be formed from the letters of the word 'MALKENKOV' such that the vowels occupy odd places, we can follow these steps:
### Step 1: Identify the letters in the word
The word 'MALKENKOV' consists of 9 letters:
- Vowels: A, A, E, O (4 vowels)
- Consonants: M, L, K, N, K, V (5 consonants)
### Step 2: Determine the positions for the vowels
In a 9-letter word, the odd positions are 1, 3, 5, 7, and 9. This gives us a total of 5 odd positions.
### Step 3: Choose positions for the vowels
Since we have 4 vowels but only 3 can occupy the 5 odd positions, we need to choose 3 out of the 5 available odd positions for the vowels. This can be calculated using combinations:
\[
\text{Number of ways to choose 3 positions from 5} = \binom{5}{3}
\]
### Step 4: Arrange the vowels
The vowels we have are A, A, E, O. Since we are using only 3 vowels (A, A, E), we need to arrange these vowels in the chosen positions. The arrangement of the vowels can be calculated using the formula for permutations of multiset:
\[
\text{Arrangements of A, A, E} = \frac{3!}{2!} = 3
\]
### Step 5: Arrange the consonants
After placing the vowels, we will have 6 positions left (2 odd positions and 4 even positions) for the consonants M, L, K, N, K, V. The consonants include a repetition of K. The number of arrangements of the consonants can be calculated as:
\[
\text{Arrangements of M, L, K, N, K, V} = \frac{6!}{2!} = 360
\]
### Step 6: Calculate the total arrangements
Now, we can calculate the total number of arrangements by multiplying the number of ways to choose the positions for the vowels, the arrangements of the vowels, and the arrangements of the consonants:
\[
\text{Total arrangements} = \binom{5}{3} \times \frac{3!}{2!} \times \frac{6!}{2!}
\]
Calculating each part:
- \(\binom{5}{3} = 10\)
- Arrangements of vowels = 3
- Arrangements of consonants = 360
Thus, the total arrangements become:
\[
\text{Total arrangements} = 10 \times 3 \times 360 = 10800
\]
### Final Answer
The total number of different words that can be formed from the letters of the word 'MALKENKOV' such that the vowels occupy odd places is **10800**.
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