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Find n from Overlapping Sets A and B Membership Counts | JEE

JEE Maths question with a full step-by-step solution.

Question

Suppose

A_1, A_2, \ldots, A_{30}

are thirty sets each having

5

elements and

B_1, B_2, \ldots, 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

A_i

and exactly

9

of the

B_j

. Then

n

is equal to:

15

3

45

correct

35

Solution

Step 1: Count memberships through the

A_i

. Total

= 30 \times 5 = 150

, and each element of

S

is counted

10

times:

10 \cdot n(S) = 150 \Rightarrow n(S) = 15.

Step 2: Count memberships through the

B_j

. Total

= 3n

, and each element of

S

is counted

9

times:

3n = 9 \cdot n(S) = 9 \times 15 = 135 \Rightarrow n = 45.

Correct answer: (3)

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