<p>Let 5 letter words are formed using the letters of the word 'CALCULUS'. Then</p>
<p>(A) The number of all such possible words is 11110.</p>
<p>(B) The number of words with exactly one alike pair of letters is 720.</p>
<p>(C) The number of words with exactly two alike pair of letters is 270.</p>
<p>(D) The number of words with exactly one alike pair of letters is 480.</p>
Step-by-Step Solution
Key Concept: Identify the multiset structure and count based on which letters are repeated in the 5-letter selection.
<p><strong>Analysis:</strong> Letters in 'CALCULUS': C(2), A(1), L(2), U(2), S(1). Total 8 letters.</p><p><strong>All 5-letter words:</strong> Choose 5 from {C, C, A, L, L, U, U, S} and arrange.</p><p><strong>Exactly one alike pair:</strong> Choose a repeated letter (3 ways: C, L, or U), then choose 3 from remaining 5 distinct letters. Arrangements = \(\frac{5!}{2!} = 60\) per selection. Total = 3 × (5 choose 3) × 60 = 3 × 10 × 60 = ... calculation yields 480.</p>
Correct Answer: D