Question

Suppose $A_1,A_2,.....................A_{3}0$ 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 $A_i$'S and exactly 9 of the $B_j$'S. Then $n$ is equal to

• A. 15
• B. 3
• C. 45
• D. 50

Answer 1

The correct option is C

45

Since S =$\bigcup _{i=1}^{30}$$A_i$

Now, element in the union S belongs to exactly 10 of the $A_i$'S. So n(S) = $\frac{1}{10}(5\times 30)$=15 Again, S =$\bigcup _{j=1}^n$$B_j$

So, n(S) = $\frac{1}{9}(3\times n)$ = $\frac{n}{3}$ Thus,$\frac{n}{3}$ = 15 $\therefore$ n = 45