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)
$30$
(B)
$50$
(C)
$15$
(D)
$45$
MEDIUM
Correct Answer:A
Explanation
$\bigcup_{i=1}^{50} X_i = X_1, X_2,....., X_{50} = 50$ sets. Given each sets having $10$ elements.
So total elements = $50 \times 10$
$\bigcup_{i=1}^{n} Y_i = Y_1, Y_2,....., Y_n = n$ sets. Given each sets having $5$ elements.
So total elements = $5 \times n$
Now each element of set $T$ contains exactly $20$ of sets $X_i$.
So number of effective elements in set $T = \frac{50 \times 10}{20}$
Also each element of set $T$ contains exactly $6$ of sets $Y_i$.
So number of effective elements in set $T = \frac{n \times 5}{6}$
$\therefore \frac{50 \times 10}{20} = \frac{n \times 5}{6}$
$\Rightarrow n = 30$