Suppose A 1 , A 2 , … , A 30 are thirty sets each with five elements and B 1 , B 2 , … , B n are ' n ' sets each with three elements. Let ? i = 1 30 A i = ? j = 1 n B j = S Assume that each element of S belongs to exactly ten of A i ' s and

Question

Suppose $A_{1}, A_{2}, \dots, A_{30}$ are thirty sets each with five elements and $B_{1}, B_{2}, \dots, B_{n}$ are $' n'$ sets each with three elements.

Let $\cup_{i = 1}^{30} A_{i} = \cup_{j = 1}^{n} B_{j} = S$

Assume that each element of $S$ belongs to exactly ten of $A_{i}' s$ and exactly 9 of $B_{j}' s$ , then the value of $n$ is

Smart Achievers · Accepted Answer

(4)

$Since each A_{i} has 5 elements, and each element of$ $S belongs to exactly 10 of the A_{i}'s$

$∴$ $S = \cup_{i = 1}^{30} A_{i} \Rightarrow n (S) = \frac{1}{10} \sum_{i = 1}^{30} n (A_{i})$

$= \frac{1}{10} \times 5 \times 30 = 15$ ...(i)

$B_{j}' s$ $exactly 9 of B_{j}'s,$

$S = \cup_{j = 1}^{n} B_{j} \Rightarrow n (S) = \frac{1}{9} \sum_{j = 1}^{n} n (B_{j})$

$= \frac{1}{9} \times 3 n = \frac{n}{3}$ ...(ii)

$From (i) and (ii) we get$

$\frac{n}{3} = 15 \Rightarrow n = 45$

Solution

1 90

2 15

3 9

4 45

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Solution

Q. Suppose A1,A2,…,A30 are thirty sets each with five elements and B1,B2,…,Bn are 'n' sets each with three elements. Let ∪i=130Ai=∪j=1nBj=S Assume that each element of S belongs to exactly ten of Ai's and exactly 9 of Bj's, then the value of n is

1 90

2 15

3 9

4 45

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