<p>If two different numbers are taken from the set \(\{0, 1, 2, 3, \ldots, 10\}\); then, the probability that their sum as well as absolute difference are both multiple of 4, is</p>
Step-by-Step Solution
Key Concept: For sum and absolute difference of two numbers to both be multiples of 4, both numbers must have the same remainder when divided by 4. Additionally, their sum being divisible by 4 requires the remainders to satisfy specific conditions (0+0, 2+2 mod 4).
<p><strong>Step 1:</strong> Partition {0,1,2,...,10} by remainders mod 4:</p><ul><li>Remainder 0: {0, 4, 8} — 3 numbers</li><li>Remainder 1: {1, 5, 9} — 3 numbers</li><li>Remainder 2: {2, 6, 10} — 3 numbers</li><li>Remainder 3: {3, 7} — 2 numbers</li></ul><p><strong>Step 2:</strong> For sum and |difference| both divisible by 4, if a ≡ r (mod 4) and b ≡ s (mod 4):</p><ul><li>Sum: a + b ≡ r + s ≡ 0 (mod 4)</li><li>Difference: |a - b| ≡ |r - s| ≡ 0 (mod 4)</li><li>This requires r ≡ s (mod 4) AND 2r ≡ 0 (mod 4)</li></ul><p><strong>Step 3:</strong> Valid cases where 2r ≡ 0 (mod 4): only r = 0 or r = 2</p><ul><li>Both from remainder 0: C(3,2) = 3 pairs</li><li>Both from remainder 2: C(3,2) = 3 pairs</li><li>Total favorable: 3 + 3 = 6 pairs</li></ul><p><strong>Step 4:</strong> Total ways to choose 2 different numbers from 11: C(11,2) = 55</p><p><strong>Step 5:</strong> Probability = 6/55</p><p>∴ Answer: D</p>
Correct Answer: D