Find the number of words that can be made out of the letters of the word MOBILE when consonants always occupy odd places.

Options

A. 20
B. 36
C. 30
D. 720

Answer

To solve the problem of finding the number of words that can be formed from the letters of the word "MOBILE" with the condition that consonants always occupy odd places, we can follow these steps:

### Step 1: Identify the letters in the word "MOBILE"
The word "MOBILE" consists of 6 letters: M, O, B, I, L, E.
- **Consonants**: M, B, L
- **Vowels**: O, I, E

### Step 2: Determine the positions for consonants and vowels
Since the word has 6 letters, the positions can be numbered as follows:
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, while the even positions are 2, 4, and 6.

### Step 3: Arrange the consonants in the odd positions
We have 3 consonants (M, B, L) and 3 odd positions (1, 3, 5). The number of ways to arrange the 3 consonants in these 3 positions is given by the factorial of the number of consonants:

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

### Step 4: Arrange the vowels in the even positions
Similarly, we have 3 vowels (O, I, E) and 3 even positions (2, 4, 6). The number of ways to arrange the 3 vowels in these 3 positions is also given by the factorial of the number of vowels:

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

### Step 5: Calculate the total arrangements
Since the arrangements of consonants and vowels are independent of each other, we can multiply the number of arrangements of consonants by the number of arrangements of vowels to get the total number of arrangements:

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

### Final Answer
Thus, the total number of words that can be formed from the letters of the word "MOBILE" with consonants occupying odd places is **36**.

---