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प्रश्न
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.
उत्तर
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.
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