<p>A forecast is to be made of the results of five cricket matches, each of which can be a win or a draw or a loss for Indian team. Let</p><p>\(p\) = number of forecasts with exactly 1 error</p><p>\(q\) = number of forecasts with exactly 3 errors</p><p>\(r\) = number of forecasts with all five errors</p><p>Then the correct statement(s) is/are</p>
Step-by-Step Solution
Key Concept: Each match has 3 possible outcomes; an 'error' means predicting the wrong outcome. Use binomial coefficients to count forecasts with exactly k errors: C(5,k) × 2^k, since for k errors, we choose k matches to get wrong and each has 2 wrong choices.
<p><strong>Step 1:</strong> For each match, there are 3 possible outcomes. An error means predicting one of the 2 wrong outcomes.</p><p><strong>Step 2:</strong> Number of forecasts with exactly k errors = C(5,k) × 2^k</p><p><strong>Step 3:</strong> Calculate p (exactly 1 error):<br>p = C(5,1) × 2¹ = 5 × 2 = 10</p><p><strong>Step 4:</strong> Calculate q (exactly 3 errors):<br>q = C(5,3) × 2³ = 10 × 8 = 80</p><p><strong>Step 5:</strong> Calculate r (all 5 errors):<br>r = C(5,5) × 2⁵ = 1 × 32 = 32</p><p><strong>Step 6:</strong> Verify relationships:<br>• p + q + r = 10 + 80 + 32 = 122 ✓<br>• q/p = 80/10 = 8 ✓<br>• r = 16p/5? → 32 = 160/5 = 32 ✓<br>• 3p + q = r? → 30 + 80 = 110 ≠ 32 ✗</p><p>∴ Answer: ACD (verify which exact statements match these values)</p>
Correct Answer: ACD