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प्रश्न
Suppose A1, A2, ..., A30 are thirty sets each having 5 elements and B1, B2, ..., Bn are n sets each with 3 elements, let \[\bigcup\limits_{i=1}^{30} A_{i} = \bigcup\limits_{j=1}^{n} B_{j}\] = and each element of S belongs to exactly 10 of the Ai’s and exactly 9 of the B,’S. then n is equal to ______.
पर्याय
15
3
45
35
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उत्तर
Suppose A1, A2, ..., A30 are thirty sets each having 5 elements and B1, B2, ..., Bn are n sets each with 3 elements, let \[\bigcup\limits_{i=1}^{30} A_{i} = \bigcup\limits_{j=1}^{n} B_{j}\] = and each element of S belongs to exactly 10 of the Ai’s and exactly 9 of the B,’S. then n is equal to 45.
Explanation:
Given: \[\bigcup\limits_{i=1}^{30} A_{i} = \bigcup\limits_{j=1}^{n} B_{j}\]
To find: value of n
Since elements are not repeating, number of elements in A1∪ A2∪ A3∪ ………∪ A30 is 30 × 5
But each element is used 10 times
So, 10 × S = 30 × 5
⇒ 10 × S = 150
⇒ S = 15
Since elements are not repeating, number of elements in B1∪ B2∪ B3∪ ………∪ Bn is 3 × n
But each element is used 9 times
So, 9 × S = 3 × n
⇒ 9 × S = 3n
⇒ S = `"n"/3`
⇒ `"n"/3` = 15
⇒ n = 45
Hence, the value of n is 45.
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