The number of permutations by taking all letters and keeping the vowels of the word COMBINE in the odd places is.

Options

A. (a)96
B. (b)144
C. (c)512
D. (d)576

Answer

To solve the problem of finding the number of permutations of the letters in the word "COMBINE" while keeping the vowels in the odd places, we can follow these steps:

### Step 1: Identify the letters and vowels
The word "COMBINE" consists of the letters: C, O, M, B, I, N, E.
The vowels in this word are O, I, and E.

### Step 2: Count the total letters and categorize them
- Total letters = 7 (C, O, M, B, I, N, E)
- Vowels = 3 (O, I, E)
- Consonants = 4 (C, M, B, N)

### Step 3: Determine the positions for the vowels
In a 7-letter arrangement, the odd positions are 1, 3, 5, and 7. Therefore, there are 4 odd positions available.

### Step 4: Select odd positions for the vowels
We need to place the 3 vowels in the 4 available odd positions. We can choose 3 positions out of 4 for the vowels. The number of ways to choose 3 positions from 4 is given by the combination formula \( C(n, r) \):
\[ C(4, 3) = \frac{4!}{3!(4-3)!} = 4 \]

### Step 5: 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 arrangements of 3 vowels is given by \( 3! \):
\[ 3! = 6 \]

### Step 6: Arrange the consonants in the remaining positions
After placing the vowels, we have 4 positions left (1 odd position remains and 3 even positions). We can arrange the 4 consonants (C, M, B, N) in these 4 positions. The number of arrangements of 4 consonants is given by \( 4! \):
\[ 4! = 24 \]

### Step 7: Calculate the total permutations
Now, we can calculate the total number of permutations by multiplying the number of ways to choose the positions for the vowels, the arrangements of the vowels, and the arrangements of the consonants:
\[ \text{Total permutations} = C(4, 3) \times 3! \times 4! = 4 \times 6 \times 24 \]
Calculating this gives:
\[ 4 \times 6 = 24 \]
\[ 24 \times 24 = 576 \]

### Final Answer
Thus, the total number of permutations by taking all letters and keeping the vowels of the word "COMBINE" in the odd places is **576**.

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