<p>Given word: BARRACK. The total number of four-letter words that can be formed using the letters of the word BARRACK is:</p>
Step-by-Step Solution
Key Concept: Break into cases based on letter repetition patterns (all different, one pair repeated, two pairs repeated) since BARRACK has repeated letters: A appears 2 times, R appears 2 times. Use the inclusion-exclusion principle or direct counting by cases.
<p><strong>Step 1:</strong> Identify available letters in BARRACK: B(1), A(2), R(2), C(1), K(1)</p><p><strong>Step 2:</strong> Case 1 - All 4 letters different: Choose 4 from {B, A, R, C, K} and arrange. C(5,4) × 4! = 5 × 24 = 120</p><p><strong>Step 3:</strong> Case 2 - Exactly one pair repeated: Either AA** or RR** where * are different letters from remaining.</p><p>For AA: Choose 2 from {B, R, C, K} and arrange with AA. C(4,2) × (4!/2!) = 6 × 12 = 72</p><p>For RR: Choose 2 from {B, A, C, K} and arrange with RR. C(4,2) × (4!/2!) = 6 × 12 = 72</p><p><strong>Step 4:</strong> Case 3 - Two pairs repeated (AARR): Arrange AARR = 4!/(2!×2!) = 6</p><p><strong>Step 5:</strong> Total = 120 + 72 + 72 + 6 = 270</p><p>∴ Answer: D</p>
Correct Answer: D