Number of ways in which the letters of the word MOBILE be arranged so that the consonants always occupy the odd places is

Options

A. 18
B. 54
C. 36
D. 72

Answer

To solve the problem of arranging the letters of the word "MOBILE" such that the consonants always occupy the odd places, we can follow these steps:

### Step 1: Identify the letters in the word "MOBILE"
The letters in "MOBILE" are: M, O, B, I, L, E.

### Step 2: Classify the letters into consonants and vowels
- **Consonants**: M, B, L (3 consonants)
- **Vowels**: O, I, E (3 vowels)

### Step 3: Determine the positions available for consonants
The word "MOBILE" has 6 letters, which means there are 6 positions:
1. Position 1 (odd)
2. Position 2 (even)
3. Position 3 (odd)
4. Position 4 (even)
5. Position 5 (odd)
6. Position 6 (even)

The odd positions are 1, 3, and 5. Therefore, there are **3 odd positions** available for the consonants.

### Step 4: Arrange the consonants in the odd positions
Since we have 3 consonants (M, B, L) and 3 odd positions, we can arrange the consonants in these positions. The number of ways to arrange 3 consonants in 3 positions is given by the factorial of the number of consonants:

\[ \text{Ways to arrange consonants} = 3! = 6 \]

### Step 5: Determine the positions available for vowels
After placing the consonants in the odd positions, the remaining positions (2, 4, and 6) will automatically be occupied by the vowels (O, I, E).

### Step 6: Arrange the vowels in the even positions
Similarly, we can arrange the 3 vowels (O, I, E) in the 3 even positions. The number of ways to arrange the vowels is also given by the factorial of the number of vowels:

\[ \text{Ways to arrange vowels} = 3! = 6 \]

### Step 7: Calculate the total arrangements
Now, to find the total number of arrangements where the consonants occupy the odd positions, we multiply the number of arrangements of consonants by the number of arrangements of vowels:

\[ \text{Total arrangements} = (\text{Ways to arrange consonants}) \times (\text{Ways to arrange vowels}) = 3! \times 3! = 6 \times 6 = 36 \]

### Final Answer
The total number of ways in which the letters of the word "MOBILE" can be arranged such that the consonants always occupy the odd places is **36**.

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