How many different ways are possible to arrange the letters of the word “MACHINE” so that the vowels may occupy only the odd positions

Options

A. 800
B. 125
C. 348
D. 576

Answer

To solve the problem of arranging the letters of the word "MACHINE" such that the vowels occupy only the odd positions, we can follow these steps:

### Step 1: Identify the letters and their types
The word "MACHINE" consists of 7 letters: M, A, C, H, I, N, E. Among these, the vowels are A, I, and E, and the consonants are M, C, H, and N.

- **Vowels**: A, I, E (3 vowels)
- **Consonants**: M, C, H, N (4 consonants)

### Step 2: Identify the positions
In a 7-letter arrangement, the odd positions are 1, 3, 5, and 7. This gives us 4 odd positions available.

### Step 3: Place the vowels in the odd positions
Since we have 3 vowels and 4 odd positions, we can choose any 3 out of the 4 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 (A, I, E) in those positions. The number of arrangements of 3 vowels is given by:

\[ 3! = 6 \]

### Step 5: Place the consonants in the remaining positions
After placing the vowels, there will be 4 remaining positions (1 odd position left and 3 even positions). We can place the 4 consonants (M, C, H, N) in these 4 positions. The number of arrangements of 4 consonants is given by:

\[ 4! = 24 \]

### Step 6: Calculate the total arrangements
Finally, we multiply the number of ways to choose the positions for the vowels, the arrangements of the vowels, and the arrangements of the consonants:

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

Calculating this gives:

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

Thus, the total number of different ways to arrange the letters of the word "MACHINE" such that the vowels occupy only the odd positions is **576**.

### Summary of Steps:
1. Identify letters and their types (vowels and consonants).
2. Identify the available odd positions.
3. Choose positions for the vowels using combinations.
4. Arrange the vowels in the chosen positions.
5. Arrange the consonants in the remaining positions.
6. Calculate the total arrangements.