# Let $\bigcup_{i=1}^{50}X_{i}=\bigcup_{i=1}^{n}Y_{i}=T$, where each Xi contains 10 elements and each Yi contains 5 elements. If each element of the set T is an element of exactly 20 sets $X_{i}'s$ and exactly 6 sets $Y_{i}'s$ then n is equal to: Option: 1 15 Option: 2 50 Option: 3 45 Option: 4 30

$\\n\left(X_{i}\right)=10\\ \bigcup_{i=1}^{50} X_{i}=T, \Rightarrow n(T)=500$

each element of T belongs to exactly 20 elements of Xi

$\Rightarrow \frac{500}{20}=25 \text { distinct element }$

$\text { so } \frac{5 \mathrm{n}}{6}=25 \Rightarrow \mathrm{n}=30$

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