Let _ i = 1 ^ 50 X _ i = _ i = 1 ^ n Y _ i = T , where each X _ i...

Let $\mathop{\cup }_{i = 1}^{50}{X}_{i} = \mathop{\cup }_{i = 1}^{n}{Y}_{i} = T$ , 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 :

Options

A $15$
B $50$
C $45$
D $30$

Correct answer

D. $30$

Step-by-step solution

$\mathop{\cup }_{i = 1}^{50}{X}_{i} = \mathop{\cup }_{i = 1}^{n}{Y}_{i} = T$

From question's conditions

$\dfrac{10 \times 50}{20} = \dfrac{5n}{6}$

$⇒ n = 30$

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