<p>The total number of three-letter words that can be formed from the letter of the word SAHARANPUR is equal to</p>
Step-by-Step Solution
Key Concept: Count three-letter words by partitioning into cases based on letter repetition patterns (all different, exactly two same, all three same), accounting for the frequency of available letters in SAHARANPUR.
<p><strong>Letter frequencies in SAHARANPUR:</strong> S(1), A(3), H(1), R(2), N(1), P(1), U(1) → 7 distinct letters</p><p><strong>Case 1: All three letters different</strong><br>Choose 3 from 7 distinct letters and arrange: P(7,3) = 7×6×5 = 210</p><p><strong>Case 2: Exactly two letters same</strong><br>Only A(3) and R(2) can be repeated.<br>• Two A's, one other: Choose 1 from {S,H,R,N,P,U} = 6 choices. Arrangements = 3!/2! = 3 per choice. Subtotal: 6×3 = 18<br>• Two R's, one other: Choose 1 from {S,A,H,N,P,U} = 6 choices. Arrangements = 3!/2! = 3 per choice. Subtotal: 6×3 = 18<br>Total for Case 2: 18 + 18 = 36</p><p><strong>Case 3: All three letters identical</strong><br>Only possible with A (since A appears 3 times). Arrangements = 1<br>Total for Case 3: 1</p><p><strong>Total = 210 + 36 + 1 = 247</strong></p><p>∴ Answer: D</p>
Correct Answer: D