How many words, with or without meaning can be formed by using all the letters of the word 'MACHINE', so that the vowels occurs only the odd position ?

Options

A. 1440
B. 720
C. 640
D. 576

Explanation

To solve the problem of how many words can be formed using all the letters of the word "MACHINE" such that the vowels occur only in odd positions, we can follow these steps:

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

### Step 2: Determine the positions for the vowels
In the word "MACHINE," we have 7 positions (1 to 7). The odd positions are 1, 3, 5, and 7. This gives us 4 odd positions.

### Step 3: Choose positions for the vowels
Since we have 3 vowels (A, I, E) and we need to place them in the 4 odd positions, we can select 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 in the chosen positions
Once we have chosen the 3 positions for the vowels, we can arrange the 3 vowels (A, I, E) in those positions. The number of ways to arrange 3 vowels is given by:

\[ 3! = 6 \]

### Step 5: Arrange the consonants in the remaining positions
After placing the vowels, we have 4 remaining positions (1 odd position left and 3 even positions). The consonants in "MACHINE" are M, C, H, and N (4 consonants). The number of ways to arrange these 4 consonants in the 4 remaining positions is:

\[ 4! = 24 \]

### Step 6: Calculate the total arrangements
Now, we can calculate the total number of arrangements by multiplying the number of ways to choose 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 words that can be formed is **576**.

### Summary of Steps:
1. Identify letters and vowels.
2. Determine odd positions.
3. Choose positions for vowels.
4. Arrange vowels in those positions.
5. Arrange consonants in remaining positions.
6. Calculate total arrangements.