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

Options

A. 96
B. 144
C. 512
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 positions, we can follow these steps:

### Step 1: Identify the letters and vowels
The word "COMBINE" consists of 7 letters: C, O, M, B, I, N, E. Among these, the vowels are O, I, and E.

### Step 2: Determine the positions for vowels
In the arrangement of 7 letters, the odd positions are 1, 3, 5, and 7. This means we have 4 odd positions available.

### Step 3: Choose positions for the vowels
We have 3 vowels (O, I, E) that need to be placed in the odd positions. We can choose 3 out of the 4 available odd positions to place the vowels. The number of ways to choose 3 positions from 4 is given by the combination formula:

\[ \text{Number of ways to choose 3 positions from 4} = \binom{4}{3} = 4 \]

### Step 4: Arrange the vowels
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:

\[ 3! = 6 \]

### Step 5: Arrange the consonants
After placing the vowels, we have 4 remaining positions (1 odd position left and 3 even positions) for the consonants C, M, B, and N. The number of arrangements of these 4 consonants is given by:

\[ 4! = 24 \]

### Step 6: 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:

\[ \text{Total permutations} = \binom{4}{3} \times 3! \times 4! = 4 \times 6 \times 24 \]

Calculating this gives:

\[ 4 \times 6 = 24 \]
\[ 24 \times 24 = 576 \]

Thus, the total number of permutations of the letters in the word "COMBINE" with the vowels in the odd positions is **576**.