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 $X_i$'s and exactly $6$ of sets $Y_i$'s, then $n$ is equal to

• A. $50$
• B. $15$
• C. $45$
• D. $30$

Answer 1

The correct option is D $30$

Given: $T=\bigcup _{i=1}^{50}{X}_i=\bigcup _{i=1}^n{Y}_i$
It is given that each set of $X$ contains $10$ elements.
$\therefore n\left(T\right)=50\times 10=500$
But it is given that each element of the set $T$ is an element of exactly $20$ of sets $X_i$'s.
$\therefore$ Number of distinct elements in $T=\frac{500}{20}\cdots \left(1\right)$

It is also given that each set of $Y$ contains $5$ elements.
$\therefore n\left(T\right)=n\times 5=5n$
But it is given that each element of the set $T$ is an element of exactly $6$ of sets $Y_i$'s.
$\therefore$ Number of distinct elements in $T=\frac{5n}{6}\cdots \left(2\right)$

From $\left(1\right)$ and $\left(2\right),$ we get
$\frac{50}{2}=\frac{5n}{6}$
$\therefore n=30$