<p>Find the probability that the 3 N's come consecutively in the arrangement of the letters of the word "CONSTANTINOPLE".</p>
Step-by-Step Solution
Key Concept: Treat the 3 N's as a single unit to find favorable outcomes, then divide by total arrangements accounting for repeated letters in the word.
<p><strong>Step 1:</strong> Count letters in CONSTANTINOPLE: C-O-N-S-T-A-N-T-I-N-O-P-L-E (14 letters)</p><p>Letter frequency: N appears 3 times, O appears 2 times, T appears 2 times, others appear once.</p><p><strong>Step 2:</strong> Total arrangements = 14!/(3!×2!×2!) [accounting for 3 N's, 2 O's, 2 T's]</p><p><strong>Step 3:</strong> For favorable outcomes, treat the 3 N's as a single block. Now we have 12 units: (NNN), C, O, S, T, A, T, I, O, P, L, E</p><p>These 12 units contain: 2 O's and 2 T's (still repeated)</p><p>Favorable arrangements = 12!/(2!×2!)</p><p><strong>Step 4:</strong> Probability = [12!/(2!×2!)] / [14!/(3!×2!×2!)]</p><p>= [12!/(2!×2!)] × [(3!×2!×2!)/14!]</p><p>= [12! × 3! × 2! × 2!] / [2! × 2! × 14!]</p><p>= [12! × 6] / [14 × 13 × 12!]</p><p>= 6/(14×13) = 6/182 = <strong>3/91</strong></p><p>∴ Answer: B</p>
Correct Answer: B