Question

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

Answer 1

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