Let _ i=1 ^ 50 X_ i = _ i=1 ^ n Y_ i =T where each X_ i contains … - Vidyadip

MCQ

Let $\bigcup \limits_{i=1}^{50} X_{i}=\bigcup \limits_{i=1}^{n} Y_{i}=T$ 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
$45$
B
$15$
C
$50$
✓
$30$

✓

Answer

Correct option: D.

$30$

d
$n \left( X _{ i }\right)=10 . \underset{ i =1}{ U } X _{ i }= T , \Rightarrow n ( T )=500$

each element of $T$ belongs to exactly 20

elements of $X _{ i } \Rightarrow \frac{500}{20}=25$ distinct elements

so $\frac{5 n}{6}=25 \Rightarrow n=30$

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