Let $\cup_{i = 1}^{50} X_{i} = \cup_{i = 1}^{n} Y_{i} = T$ , where each $X_{i}$ contains 10 elements and each $Y_{i}$ contains 5 elements. If each element of the set T is an element of exactly 20 of sets $X_{i}$ s and exactly 6 of sets $Y_{i}$ 's then the value of $n$ is equal to

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Detailed Solution

$\cup_{i = 1}^{50} X_{i} = \cup_{i = 1}^{n} Y_{i} = T$ ; $∵ n (X_{i}) = 10, n (Y_{i}) = 5$
So, $\cup_{i = 1}^{50} X_{i} = 500, \cup_{i = 1}^{n} Y_{i} = 5 n \Rightarrow \frac{500}{20} = \frac{5 n}{6} \Rightarrow n = 30$

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