Arrangement of Letters in the Word MOBILE

To solve this problem, we need to apply the permutation and combination formulae. We know that the word MOBILE consists of six letters, out of which there are two vowels (O, I) and four consonants (M, B, L, E).

Step 1: Fix the positions of vowels

To ensure that the consonants always occupy the odd places, we need to fix the positions of the vowels (which are the only even places). There are two ways to do this:

- The first vowel (O) can be placed in any of the three even places (2nd, 4th, or 6th), and the second vowel (I) can be placed in any of the two remaining even places.
- The first vowel (I) can be placed in any of the three even places (2nd, 4th, or 6th), and the second vowel (O) can be placed in any of the two remaining even places.

Therefore, the total number of ways to fix the positions of the vowels is:

2 × 3 × 2 = 12

Step 2: Arrange the consonants in the remaining odd places

Once we have fixed the positions of the vowels, we need to arrange the consonants (which are the only remaining letters) in the odd places. There are four consonants, and four odd places available. Therefore, the number of ways to arrange the consonants in the odd places is:

4! = 24

Step 3: Multiply the results

Finally, we need to multiply the results of Step 1 and Step 2 to get the total number of ways to arrange the letters of the word MOBILE such that consonants always occupy the odd places. Therefore, the answer is:

12 × 24 = 288

However, we need to divide this result by 2 to eliminate the double-counting of arrangements in which the vowels are swapped. Therefore, the final answer is:

288 ÷ 2 = 144

Therefore, the correct option is (A) 36.

Consonants : o,I,e

the arrangement

frist we arrange consonants in odd places in 3p1,2p1,1p1 ways

remaining in 3p3 ways

3*2*1*6=36