Let , where each contains 10 elements and each contains 5 elements. If each element of the set T is an element of exactly 20 of sets 's and exactly 6 of sets 's, then is equal to

Question

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 $n$ is equal to

Smart Achievers · Accepted Answer

(4)

$\cup_{i = 1}^{50} X_{i} = \cup_{i = 1}^{50} Y_{i} = T$

$∵$ $n (X_{i}) = 10, n (Y_{i}) = 5$

$So, ∪_{i = 1}^{50} X_{i} = 500,$ $\cup_{i = 1}^{50} Y_{i} = 5 n$

$\Rightarrow \frac{500}{20} = \frac{5 n}{6}$

$\Rightarrow n = 30$

Solution

Q.
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 $n$ is equal to

1 15

2 50

3 45

4 30

Ans.
(4)

$\cup_{i = 1}^{50} X_{i} = \cup_{i = 1}^{50} Y_{i} = T$

$∵$ $n (X_{i}) = 10, n (Y_{i}) = 5$

$So, ∪_{i = 1}^{50} X_{i} = 500,$ $\cup_{i = 1}^{50} Y_{i} = 5 n$

$\Rightarrow \frac{500}{20} = \frac{5 n}{6}$

$\Rightarrow n = 30$

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