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  • Suppose A1, A2, …, A30 are thirty sets each having 5 elements and B1, B2, …, Bn are n sets each with 3 elements, let 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 A. 15 B. 3 C. 45 D. 35

 Suppose A_1, A_2, ..., A_{30} are thirty sets each having 5 elements and B_1, B_2, ..., B_{n} are n sets each with 3 elements, let 

\bigcup_{i=1}^{30} A_i = \bigcup_{j=1}^{n}B_j = S

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
A. 15
B. 3
C. 45
D. 35
 

Answers (1)

Number of elements in \begin{aligned}A_1\cup A_2\cup A_3\cup...\cup A_{30} = 30\times 5 = 150\\ (\text{when repetition is allowed}) \end{aligned}

However, each element is repeated 10 times.

n(S) = 30 \times \frac{5}{10} = \frac{150}{10} = 15\quad (i)

Number of elements in B_1\cup B_2\cup B_3\cup...\cup B_n = 3n \ (\text{When repetition is not allowed})

But each element is repeated 09 times

        n(S) = \frac{3n}{9} = \frac{n}{3}

From (i) and (ii) we get \frac{n}{3} = 15; n = 45. hence, the correct option is (c)

 

 

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