<p>\(|X|=10\). Subsets \(A\) and \(B\) of \(X\) are chosen independently and at random (non-empty). Which of the following equal \(P(|A|=|B|)\)? <em>[JEE Advanced 2017]</em></p>
Step-by-Step Solution
Key Concept: Total subsets (including empty): 2^10 each. P(|A|=|B|) = \Sigma[C(10,k)/2^10]^2 = C(20,10)/4^10. Options A and D are consistent.
<p>Total subsets of X: \(2^{10}\) (including empty). Choose A and B independently.</p><p>\(P(|A|=k) = \dfrac{\binom{10}{k}}{2^{10}}\) for each \(k=0,1,\ldots,10\).</p><p>\(P(|A|=|B|) = \sum_{k=0}^{10}\left[\dfrac{\binom{10}{k}}{2^{10}}\right]^2 = \dfrac{\sum_k \binom{10}{k}^2}{4^{10}} = \dfrac{\binom{20}{10}}{4^{10}}\)</p><p>using Vandermonde identity \(\sum_k\binom{n}{k}^2 = \binom{2n}{n}\). ✓ Options A and D both state this. Answer: AD ✓</p>
Correct Answer: AD