Suppose A1-A2-A30 are thirty sets each having 5 elements and B1-B2-Bn are n sets each with 3 elements- Let - i -130 Ai - Uj -1 nBj - S and each element of S belong to exactly 10 of the Ai-s and exactly 9 of the Bj-s- then n is equal toa 15b 3c 45d 35

It is given that each set Ai 1 #8804;i #8804;30 contains 5 elements and #8746;i=130Ai=S. #8756;nS=30 #215;5=150 But, it is given that each element ...

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Standard XII

Mathematics

Union

Suppose A1,A2...

Question

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 $\cup_{i = 1}^{30} A_{i} = \cup_{j = 1}^{n} B_{j} = S$ and each element of S belong to exactly 10 of the $A_{i}' s$ and exactly 9 of the $B_{j}' s$ , then n is equal to

(a) 15 (b) 3 (c) 45 (d) 35

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Solution

It is given that each set A_i $(1 \leq i \leq 30)$ contains 5 elements and $\cup_{i = 1}^{30} A_{i} = S$ .

$∴ n (S) = 30 \times 5 = 150$

But, it is given that each element of S belong to exactly 10 of the A_i's.

∴ Number of distinct elements in S = $\frac{150}{10} = 15$ .....(1)

It is also given that each set B_j $(1 \leq j \leq n)$ contains 3 elements and $\cup_{j = 1}^{n} B_{j} = S$ .

$∴ n (S) = n \times 3 = 3 n$

Also, each element of S belong to eactly 9 of B_j's.

∴ Number of distinct elements in S = $\frac{3 n}{9}$ .....(2)

From (1) and (2), we have

$\frac{3 n}{9} = 15 \Rightarrow n = 45$

Thus, the value of n is 45.

Hence, the correct answer is option (c).

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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 ∪i=130Ai=∪j=1nBj=S and each element of S belong to exactly 10 of the Ai's and exactly 9 of the Bj's, then n is equal to (a) 15 (b) 3 (c) 45 (d) 35