Each set $X$, contains 5 elements and each set $Y$, contains 2 elements and $\bigcup_{r=1}^{20} X_{r}=S=\bigcup_{r=1}^{n} Y_{r}$. If each element of $S$ belong to exactly 10 of the $X_{r}^{\prime} s$ and to eactly 4 of $Y_{r}^{\prime} s$, then find the value of $n$.
It is given that each set $X$ contains 5 elements and $\bigcup_{r=1}^{20} X_{r}=S$.
$\therefore n(S)=20 \times 5=100$
But, it is given that each element of $S$ belong to exactly 10 of the $X_{r}$ 's.
$\therefore$ Number of distinct elements in $S=\frac{100}{10}=10$ n...91)
It is also given that each set $Y$ contains 2 elements and $\bigcup_{r=1}^{n} Y_{r}=S$.
$\therefore n(S)=n \times 2=2 n$
Also, each element of $S$ belong to eactly 4 of $Y_{r}$ 's.
$\therefore$ Number of distinct elements in $S=\frac{2 n}{4}$ ...(2)
From (1) and (2), we have
$\frac{2 n}{4}=10$
$\Rightarrow n=20$
Hence, the value of $n$ is 20 .