Each set X_r contains 5 elements and each set Y_r contains 4 elem… - Vidyadip

MCQ

Each set $X_r$ contains $5$ elements and each set $Y_r$ contains $4$ elements and
$\bigcup\limits_{r = 1}^{24} {{X_r} = S = \bigcup\limits_{r = 1}^n {{Y_r}} }$ If each element of set $S$ belong to exactly $10$ of the $X_r's$ and to exactly $6$ of $Y_r's$, then $n$ is (where $\bigcup\limits_{r = 1}^{24} {X_r}$ denotes $X_1 \cup X_2 \cup X_3 \cup ....... \cup X_{24})$

✓
$18$
B
$15$
C
$20$
D
$24$

✓

Answer

Correct option: A.

$18$

a
Total elements (with repeation) in

$\bigcup\limits_{r = 1}^{24} {{{\rm{X}}_{\rm{r}}} = 5 \times 24 = 120} $

but each element lie in $10{\rm{ }}{{\rm{x}}_r}$ 's

$\Rightarrow$ Total different elements $=\frac{120}{10}=12$ ....$(i)$

Similarly from $\bigcup\limits_{r = 1}^n {{{\rm{Y}}_{\rm{r}}}} $

we get total different elements $=\frac{4 \mathrm{n}}{6}$ .....$(ii)$

from $(i)$ and $(ii),$ $12=\frac{4 \mathrm{n}}{6}$

$ \Rightarrow \mathrm{n}=18$

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