<p>Find the number of ways in which India can win the series of 11 matches (If no match is drawn and all matches are played).</p>
Step-by-Step Solution
Key Concept: India must win at least 6 out of 11 matches to clinch the series (majority); sum the binomial coefficients C(11,6) + C(11,7) + ... + C(11,11), which equals 2^10 by the symmetry property of binomial coefficients.
<p><strong>Step 1:</strong> For India to win a series of 11 matches, India must win more than half = at least 6 matches.</p><p><strong>Step 2:</strong> Number of ways = C(11,6) + C(11,7) + C(11,8) + C(11,9) + C(11,10) + C(11,11)</p><p><strong>Step 3:</strong> By symmetry of binomial coefficients: Σ C(11,k) for k=0 to 11 equals 2^11. Also, Σ C(11,k) for k=0 to 5 = Σ C(11,k) for k=6 to 11 (by symmetry C(11,k) = C(11,11-k)).</p><p><strong>Step 4:</strong> Since total outcomes = 2^11, the ways India wins = ways India loses = 2^11/2 = 2^10</p><p>∴ Answer: <strong>2^10</strong> or <strong>1024</strong></p>
Correct Answer: 2^10