<p>A person predicts the outcome of 20 cricket matches of his home team. Each match can result in either a win, loss, or tie for the home team. Total number of ways in which he can make the predictions so that exactly 10 predictions are correct is equal to</p>
<p>\({}^{20}C_{10} \times 2^{10}\)</p>
<p>\({}^{20}C_{10} \times 3^{20}\)</p>
<p>\({}^{20}C_{10} \times 3^{10}\)</p>
<p>\({}^{20}C_{10} \times 2^{20}\)</p>
Step-by-Step Solution
Key Concept: The person makes 20 predictions (each W/L/T), and we want exactly 10 to be correct. For any fixed actual outcome sequence, the number of prediction sequences with exactly 10 matches correct is C(20,10)×2^10, since we choose 10 positions to predict correctly and for the remaining 10 positions, each must be one of the 2 wrong outcomes.
<p><strong>Step 1:</strong> Identify what we're counting. The person makes 20 predictions, each being one of {W, L, T}. We want to count total prediction sequences where exactly 10 match the (unknown) actual outcomes.</p><p><strong>Step 2:</strong> For a fixed sequence of actual match results, the number of prediction sequences with exactly 10 correct is: choose which 10 of 20 predictions are correct: C(20,10) ways.</p><p><strong>Step 3:</strong> For each of the remaining 10 positions, the prediction must be wrong. If the actual outcome is one specific result, there are exactly 2 wrong predictions available.</p><p><strong>Step 4:</strong> Therefore, for the 10 incorrect predictions, there are 2^10 ways to assign the wrong outcomes.</p><p><strong>Step 5:</strong> Total ways = C(20,10) × 2^10</p><p>∴ Answer: <strong>C(20,10) × 2^10</strong> or equivalently <strong>C(20,10) × 1024</strong></p>
Correct Answer: A