<p>Four numbers are chosen at random (without replacement) from the set \(\{1, 2, 3, \ldots, 20\}\).<br><strong>Statement-1:</strong> The probability that the chosen numbers when arranged in some order will form an AP is \(1/85\).<br><strong>Statement-2:</strong> If the four chosen numbers from an AP, then the set of all possible values of common difference is \(\{\pm1, \pm2, \pm3, \pm4, \pm5\}\).</p>
<p>Statement-1 is true, Statement-2 is false.</p>
<p>Statement-1 is true, Statement-2 is true; Statement-2 is the correct explanation for Statement-1.</p>
<p>Statement-1 is true, Statement-2 is true; Statement-2 is not the correct explanation for Statement-1.</p>
<p>Statement-1 is false, Statement-2 is true.</p>
Step-by-Step Solution
Key Concept: To find the probability, count all 4-term APs in {1,2,...,20} by fixing common difference d, then divide by total ways C(20,4). For a 4-term AP with first term a and common difference d: we need a, a+d, a+2d, a+3d all in [1,20], giving constraints on both a and d.
<p><strong>Step 1: Count total ways to choose 4 numbers from 20</strong></p><p>Total outcomes = C(20,4) = 4845</p><p><strong>Step 2: Count 4-term APs by common difference d</strong></p><p>For AP: a, a+d, a+2d, a+3d where d ≥ 1 (consider only positive d, then account for direction)</p><p>Need: 1 ≤ a and a+3d ≤ 20, so a ≤ 20-3d</p><p>• d=1: a ∈ {1,2,...,17} → 17 APs</p><p>• d=2: a ∈ {1,2,...,14} → 14 APs</p><p>• d=3: a ∈ {1,2,...,11} → 11 APs</p><p>• d=4: a ∈ {1,2,...,8} → 8 APs</p><p>• d=5: a ∈ {1,2,...,5} → 5 APs</p><p>• d=6: a ∈ {1,2} → 2 APs</p><p>Total = 17+14+11+8+5+2 = 57 APs</p><p><strong>Step 3: Verify Statement-2</strong></p><p>For d=6: a+3(6)=a+18 ≤ 20 gives a ≤ 2 ✓</p><p>For d=7: a+21 ≤ 20 is impossible ✗</p><p>So possible values of |d| are {1,2,3,4,5,6}, but Statement-2 claims {1,2,3,4,5} — this is FALSE</p><p><strong>Step 4: Calculate probability</strong></p><p>Probability = 57/4845 = 19/1615 ≈ 0.01176... ≠ 1/85 ≈ 0.01176</p><p>Actually 1/85 = 57/4845 ✓</p><p><strong>Statement-1: TRUE</strong> (probability is 1/85)</p><p><strong>Statement-2: FALSE</strong> (d can be ±6 also, not just ±1,±2,±3,±4,±5)</p><p>∴ Answer: A</p>
Correct Answer: A