# Each Set X, Contains 5 Elements and Each Set Y, Contains 2 Elements and ∪ 20 R = 1 X R = S = ∪ N R = 1 Y R If Each Element of S Belong to Exactly 10 of the Xr'S and to Eact - Mathematics

Each set X, contains 5 elements and each set Y, contains 2 elements and $\cup^{20}_{r = 1} X_r = S = \cup^n_{r = 1} Y_r$ If each element of S belong to exactly 10 of the Xr's and to eactly 4 of Yr's, then find the value of n.

#### Solution

It is given that each set X contains 5 elements and $\cup^{20}_{r = 1} X_r = S$

$\therefore n\left( S \right) = 20 \times 5 = 100$

But, it is given that each element of S belong to exactly 10 of the Xr's.
∴ Number of distinct elements in S =$\frac{100}{10} = 10$  .....(1)

It is also given that each set Y contains 2 elements and $\cup^n_{r = 1} Y_r = S$

$\therefore n\left( S \right) = n \times 2 = 2n$

Also, each element of S belong to eactly 4 of Yr's.

∴ Number of distinct elements in S = $\frac{2n}{4}$   .....(2)

From (1) and (2), we have

$\frac{2n}{4} = 10$
$\Rightarrow n = 20$

Hence, the value of n is 20.

Concept: Universal Set
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#### APPEARS IN

RD Sharma Class 11 Mathematics Textbook
Chapter 1 Sets
Exercise 1.6 | Q 15 | Page 27