Question

Each set X_r contains 5 elements and each set Y_r contains 2 elements and $\underset{r=1}{\overset{20}{\cup }}{X}_r=S=\underset{r=1}{\overset{n}{\cup }}{Y}_r$. If each element of S belongs to exactly 10 of the $X_r^{\text{'}s}$ and to exactly 4 of the $Y_r^{\text{'}s},$ then n is
(a) 10
(b) 20
(c) 100
(d) 50

Answer 1

Let us suppose
Each x_r contains 5 elements and each y_r contains 2 elements such that $\underset{r=1}{\overset{20}{\cup }}{X}_r=S=\underset{r=1}{\overset{n}{\cup }}{Y}_r$
∴ n(S) = 20 × 5 (∵ each x_r has 5 elements)
n(S) = 100

It is given that each element of 5 belong to exactly 10 of the x_r's.
∴ Number of distinct elements in $S=\frac{100}{10}=10$
Since each y_r has 2 elements and $\underset{r=1}{\overset{u}{\cup }}{Y}_r=S$
∴ n(S) = n × 2 = 2n
And each element of S belong to exactly 4 of y_r's
⇒ number of distinct elements in $S=\frac{2n}{4}\left(2\right)$
from (1) and (2)
$$\begin{gathered} 10=\frac{2n}{4} \\ \mathrm{i}.\mathrm{e}n=20 \end{gathered}$$

Hence, the correct answer is option B.