# Prove the following

Question:

Let $\bigcup_{i=1}^{50} X_{i}=\bigcup_{i=1}^{n} Y_{i}=T$, 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 $X_{i}^{\prime}$ s and exactly 6 of sets $Y_{i}^{\prime}$ s, then $n$ is equal to

1. (1) 15

2. (2) 50

3. (3) 45

4. (4) 30

Correct Option: , 4

Solution:

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

$\because \quad n\left(X_{i}\right)=10, n\left(Y_{i}\right)=5$

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

$\Rightarrow \frac{500}{20}=\frac{5 n}{6} \Rightarrow n=30$