To solve the problem of arranging the letters of the word 'COMBINE' such that the vowels occupy only the odd places, we can follow these steps:
### Step 1: Identify the vowels and consonants
The word 'COMBINE' consists of 7 letters:
- Vowels: O, I, E (3 vowels)
- Consonants: C, M, B, N (4 consonants)
### Step 2: Determine the positions available for vowels
In a 7-letter arrangement, the odd positions are:
1. Position 1
2. Position 3
3. Position 5
4. Position 7
This gives us a total of 4 odd positions.
### Step 3: Choose positions for the vowels
We need to choose 3 out of the 4 available odd positions for the vowels. The number of ways to choose 3 positions from 4 can be calculated using the combination formula \( C(n, r) \):
\[
C(4, 3) = \frac{4!}{3!(4-3)!} = \frac{4!}{3! \cdot 1!} = 4
\]
### Step 4: Arrange the vowels in the chosen positions
Once we have chosen the positions for the vowels, we can arrange the 3 vowels (O, I, E) in those positions. The number of ways to arrange 3 vowels is given by the permutation formula \( P(n) = n! \):
\[
3! = 6
\]
### Step 5: Arrange the consonants in the remaining positions
After placing the vowels, we have 4 remaining positions (which are the odd position not chosen and the 4 even positions) for the consonants (C, M, B, N). The number of ways to arrange these 4 consonants is:
\[
4! = 24
\]
### Step 6: Calculate the total arrangements
Now, we can calculate the total 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:
\[
\text{Total arrangements} = C(4, 3) \times 3! \times 4! = 4 \times 6 \times 24
\]
Calculating this gives:
\[
4 \times 6 = 24
\]
\[
24 \times 24 = 576
\]
### Final Answer
Thus, the total number of ways to arrange the letters of the word 'COMBINE' such that the vowels occupy only the odd places is **576**.
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