Question

Let $\underset{i=1}{\stackrel{50}{U}}{X}_i=\underset{i=1}{\stackrel{n}{U}}{Y}_i=T$ 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)

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