Let A1,A2,...,A40 be 40 sets, each with 7 elements and B1,B2,...Bn be n sets, each with 4 elements. If U(i=1 to 40) Ai =U(j=1 to 40) Bj=S and each element of S belongs to exactly 10 of Ais and exactly 7 of Bjs , then n is equal to_______

The number of elements in the union of the A sets is:5(30)−rAwhere r is the number of repeats.Likewise the number of elements in the B sets is:3n−rB

Each element in the union (in S) is repeated 10 times in A, which means if x was the real number of elements in A (not counting repeats) then 9 out of those 10 should be thrown away, or 9x.  Likewise on the B side, 8x of those elements should be thrown away. so now we have:150−9x=3n−8x⟺150−x=3n⟺50−x3=n

Now, to figure out what x is, we need to use the fact that the union of a group of sets contains every member of each set.  if every element in S is repeated 10 times, that means every element in the union of the A's is repeated 10 times.  This means that:150 /10=15is the number of elements in the the A's without repeats counted (same for the Bs as well).So now we have:50−15 /3=n⟺n=45