Each set X, contains 5 elements and each set Y,
Question:

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 $n$.

Solution:

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 .

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