Solve this following


Let $\bigcup_{\mathrm{i}=1}^{50} \mathrm{X}_{\mathrm{i}}=\mathrm{U}_{\mathrm{i}=1}^{\mathrm{n}} \mathrm{Y}_{\mathrm{i}}=\mathrm{T}$, where each $\mathrm{X}_{\mathrm{i}}$ contains 10

elements and each $Y_{i}$ contains 5 elements. If each element of the set $\mathrm{T}$ is an element of exactly 20 of sets $X_{i}{ }^{\prime} s$ and exactly 6 of sets $Y_{i}{ }^{\prime} s$, then $\mathrm{n}$ is equal to :


  1. 45

  2. 15

  3. 50

  4. 30

Correct Option: , 4


$\mathrm{n}\left(\mathrm{X}_{\mathrm{i}}\right)=10 . \quad \mathrm{U}_{\mathrm{i}=1}^{50} \mathrm{X}_{\mathrm{i}}=\mathrm{T}, \Rightarrow \mathrm{n}(\mathrm{T})=500$

each element of $\mathrm{T}$ belongs to exactly 20

elements of $\mathrm{X}_{\mathrm{i}} \Rightarrow \frac{500}{20}=25$ distinct elements

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


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