Question

Suppose $A_1, A_2, ....., A_{30}$ are 30 sets, each with 5 elements and $B_1, B_2, ....., B_n$ are n sets, each with 3 elements. $\bigcup_{i=1}^{30} A_i = \bigcup_{j=1}^{n} B_j = S$, and each element of S belongs to exactly 10 of $A_i$'s and 9 of $B_j$'s. Find 'n'.

• A. 90
• B. 15
• C. 9
• D. 45

Answer 1

Answer: (D)

45

Answer 2

Total occurrences in A-sets = $30 \times 5 = 150$. Since each element appears in 10 A-sets, the number of distinct elements is

$$|S| = \frac{150}{10} = 15.$$

Similarly, total occurrences in B-sets = $3n$. Since each element appears in 9 B-sets,

$$|S| = \frac{3n}{9} = \frac{n}{3}.$$

Equate both expressions:

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