<p>Find the number of words that can be made out of the letters of the word MOBILE when consonants always occupy odd places.</p>
Step-by-Step Solution
Key Concept: Identify the fixed constraint first: consonants (M, B, L) must occupy only odd positions (1st, 3rd, 5th), while vowels (O, I, E) fill the remaining even positions (2nd, 4th, 6th). The number of arrangements is the product of permutations in each group.
<p><strong>Step 1: Identify letters in MOBILE</strong></p><p>Consonants: M, B, L (3 consonants)<br>Vowels: O, I, E (3 vowels)<br>Total: 6 letters</p><p><strong>Step 2: Identify positions</strong></p><p>In a 6-letter word, odd positions are: 1st, 3rd, 5th (3 positions)<br>Even positions are: 2nd, 4th, 6th (3 positions)</p><p><strong>Step 3: Apply constraint</strong></p><p>Consonants must occupy odd positions (1st, 3rd, 5th)<br>Vowels must occupy even positions (2nd, 4th, 6th)</p><p><strong>Step 4: Count arrangements</strong></p><p>Number of ways to arrange 3 consonants in 3 odd positions = 3! = 6<br>Number of ways to arrange 3 vowels in 3 even positions = 3! = 6</p><p><strong>Step 5: Apply multiplication principle</strong></p><p>Total number of words = 3! × 3! = 6 × 6 = 36</p><p>∴ Answer: <strong>36</strong></p>
Correct Answer: 36