Question

Suppose A. A A, are thirty sets each having 5 elements and B, B.,B.
are n sets each with 3 elements, let UA - ÜB, = S and each element of S
belongs to exactly 10 of the A's and exactly 9 of the B.'S. then n is equal to
(A) 15
(B) 3
(C) 45
(D) 35
The number of subsets of the first set is​

Answer 1

Since, each A

i

has 5 elements, we have

∑

i=1

30

n(A

i

)=5×30=150 ...(1)

Let set S consists of k elements.

Since, each elements in S belongs to exactly 10 of A

i

's

Therefore,

∑

i=1

30

n(A

i

)=10k ...(2)

Hence, from (1) and (2)

10k=150⇒k=15

Now, since each B

j

has 3 elements and each S belongs to exactly 9 of B

j

's

Therefore,

∑

j=1

n

n(B

j

)=3n and ∑

j=1

n

n(B

j

)=9k

⇒3n=9k=9×15

⇒n=45