<p>If all the words formed from the letters of the word HORROR are arranged in the opposite order as they are in a dictionary, then the rank of the word HORROR is</p>
Step-by-Step Solution
Key Concept: To find the rank of HORROR when words are arranged in reverse alphabetical order, count how many words come AFTER HORROR in dictionary order (these appear BEFORE it in reverse order), then add 1.
<p><strong>Step 1:</strong> Identify the letters in HORROR: H, O, R, R, O, R (Total: 6 letters with H:1, O:2, R:3)</p><p><strong>Step 2:</strong> In reverse alphabetical order (Z to A), we need to count words that come AFTER HORROR in normal dictionary order.</p><p><strong>Step 3:</strong> Total arrangements of HORROR = 6!/(1!·2!·3!) = 720/12 = 60</p><p><strong>Step 4:</strong> Count words that come after HORROR in normal alphabetical order:</p><p>• Words starting with R: Fix R in position 1. Remaining letters: H, O, R, O, R (5 letters with H:1, O:2, R:2)<br/>Arrangements = 5!/(1!·2!·2!) = 120/4 = 30</p><p><strong>Step 5:</strong> Words starting with O: Already come before HORROR in normal order</p><p><strong>Step 6:</strong> Words starting with H:</p><p>• HO...: Position 2 is O. Remaining: R, R, O, R gives 4!/(3!·1!) = 4 arrangements (all come after HO in some sense, need careful count)</p><p>• HR...: Position 2 is R. This is lexicographically after HO, giving 4!/(1!·2!·1!) = 12 arrangements</p><p><strong>Step 7:</strong> Words starting with HR and coming after HORROR: Remaining H,O,R,O,R must form ...O,R,R,O or later = 2 arrangements come after HORROO = 1 word (HORROR itself) plus words after it = 1</p><p><strong>Step 8:</strong> Total words after HORROR in normal order = 30 + 1 = 31</p><p><strong>Step 9:</strong> In reverse alphabetical order, rank of HORROR = 31 + 1 = <strong>32</strong></p><p>∴ Answer: C (Rank = 32)</p>
Correct Answer: C