In how many ways can the letters of the word MOBILE be so that the consonants always occupy the odd places?

Options

A. A. 18
B. B. 36
C. C. 54
D. 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 consonants and vowels
The word "MOBILE" consists of 6 letters:
- Consonants: M, B, L (3 consonants)
- Vowels: O, I, E (3 vowels)

### Step 2: Determine the positions for consonants
In a 6-letter word, the odd positions are 1, 3, and 5. Therefore, we need to place the 3 consonants in these 3 odd positions.

### Step 3: Arrange the consonants
The number of ways to arrange the 3 consonants (M, B, L) in the 3 odd positions can be calculated using the factorial of the number of consonants:
- Number of arrangements = 3! = 3 × 2 × 1 = 6 ways

### Step 4: Determine the positions for vowels
The even positions in the 6-letter word are 2, 4, and 6. We will place the 3 vowels (O, I, E) in these positions.

### Step 5: Arrange the vowels
The number of ways to arrange the 3 vowels (O, I, E) in the 3 even positions can also be calculated using the factorial of the number of vowels:
- Number of arrangements = 3! = 3 × 2 × 1 = 6 ways

### Step 6: Calculate the total arrangements
The total number of arrangements of the letters in the word "MOBILE" with consonants in odd positions and vowels in even positions can be calculated by multiplying the arrangements of consonants and vowels:
- Total arrangements = (Arrangements of consonants) × (Arrangements of vowels) = 6 × 6 = 36 ways

### Final Answer
Thus, the total number of ways to arrange the letters of the word "MOBILE" such that the consonants always occupy the odd places is **36**.

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