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Question
Each set Xr contains 5 elements and each set Yr 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 Xr’s and to exactly 4 of the Yr’s, then n is ______.
Options
10
20
100
50
Solution
Each set Xr contains 5 elements and each set Yr 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 Xr’s and to exactly 4 of the Yr’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 Yr’s and each Yr contain 2 elements.
If S has n number of Yr in it.
Then `(2"n")/4` = 10
Which gives n = 20
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