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Each set Xr contains 5 elements and each set Yr contains 2 elements and ⋃r=120Xr=S=⋃r=1nYr If each element of S belong to exactly 10 of the Xr’s and to exactly 4 of the Yr’s, then n is ______. - Mathematics

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MCQ
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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

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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

Concept: Operations on Sets - Union Set
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APPEARS IN

NCERT Mathematics Exemplar Class 11
Chapter 1 Sets
Solved Examples | Q 14 | Page 11

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