Question

Let $\underset{i = 1}{\overset{50}{U}}\,X_{i} = \underset{i=1}{\overset{n}{U}} \,Y_{i} = T$ 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 :

• A. 45
• B. 15
• C. 50
• D. 30

Answer 1

Answer: (D)

$n ( X _{ i })=10 .\underset{i=1}{\overset{50}{U}}= T$ ,
$\Rightarrow n ( T )=500 $
each element of $T$ belongs to exactly $20$
elements of $X_{i} \Rightarrow \frac{500}{20}=25$ distinct elements
so $\frac{5 n}{6}=25$
$\Rightarrow n=30$