Each set X_{r} contains 5 elements and each set Y_{r} contains 2 elements and \[\bigcup\limits_{r=1}^{20} X_{r} = S = \bigcup\limits_{r=1}^{n} Y_{r}\] If each element of S belong to exactly 10 of the X_{r}’s and to exactly 4 of the Y_{r}’s, then n is ______.

#### Options

10

20

100

50

#### Solution

Each set X_{r} contains 5 elements and each set Y_{r} contains 2 elements and \[\bigcup\limits_{r=1}^{20} X_{r} = S = \bigcup\limits_{r=1}^{n} Y_{r}\] If each element of S belong to exactly 10 of the X_{r}’s and to exactly 4 of the Y_{r}’s, then n is **20**.

**Explanation:**

Since, `"n"("X"_"r")` = 5

\[\bigcup\limits_{r=1}^{20} X\] = S

We get n(S) = 100

But each element of S belong to exactly 10 of the `"X"_"r"`'s

So, `100/10` = 10 are the number of distinct elements in S.

Also each element of S belong to exactly 4 of the Y_{r}’s and each Y_{r} contain 2 elements.

If S has n number of Y_{r} in it.

Then `(2"n")/4` = 10

Which gives n = 20