<p>The number of words which can be made from letters of the word INTERMEDIATE is</p>
<p>(a) 907200 if words start with I and end with E</p>
<p>(b) 21600 if vowels and consonants occupy their original places</p>
<p>(c) 43200 if vowels and consonants occur alternatively</p>
<p>(d) 302400 if all the vowels occur together.</p>
Step-by-Step Solution
Key Concept: Count total letters and identify repeated letters carefully. Use the formula n!/(n₁!×n₂!×...×nₖ!) where nᵢ are frequencies of repeated letters.
<p><strong>Step 1:</strong> Count the letters in INTERMEDIATE.</p><p>I-N-T-E-R-M-E-D-I-A-T-E has 12 letters total.</p><p><strong>Step 2:</strong> Identify repeated letters and their frequencies.</p><p>I appears 2 times, E appears 3 times, T appears 2 times, and all others (N, R, M, D, A) appear 1 time each.</p><p><strong>Step 3:</strong> Apply the formula for permutations with repetition.</p><p>Number of words = 12!/(2!×3!×2!) = 479,001,600/(2×6×2) = 479,001,600/24 = 19,958,400</p><p>∴ Answer: A</p>
Correct Answer: A