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 :

$15$

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$30$

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$50$

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$45$

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Solution

The correct option is B
$30$

Explanation for the correct option:
Number of elements of set $T_{} = \cup_{i = 1}^{50} X_{i} = \cup_{i = 1}^{n} Y_{i}$
$\Rightarrow 50 \times \frac{10}{20} = n \times \frac{5}{6} \Rightarrow n = (\frac{50}{2}) \times (\frac{6}{5}) n = 10 \times 3 n = 30$
Hence, the correct option is (B).

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Similar questions

Q. Each set X_r contains 5 elements and each set Y_r contains 2 elements and

\cup_{r = 1}^{20} X_{r} = S = \cup_{r = 1}^{n} Y_{r}

. If each element of S belongs to exactly 10 of the

X_{r}^{' s}

and to exactly 4 of the

Y_{r}^{' s},

then n is
(a) 10
(b) 20
(c) 100
(d) 50

Each set X, contains 5 elements and each set Y, contains 2 elements and $⋃_{r = 2}^{20} X_{r} = S = ⋃_{r = 1}^{n} Y_{r}$ . If each element of S belongs to exactly 10 of the $X_{r}^{^{'} s}$ and to exactly 4 of $Y_{r}^{^{'} s}$ then find the value of n.