Question

There are four large groups of people, each with 1000 members. any two of these groups have 100 members in common. any three of these groups have 10 members in common. and there is 1 person in all four groups. all together, how many people are in these groups? (scheinerman 116) scheinerman, edward

a. mathematics: a discrete introduction, 3rd edition. cengage learning, 20120305. vitalbook file.

Answer 1

Let the groups be A, B, C and D. Then

n(A ∩ B ∩ C ∩ D) = 1
n(A ∩ B ∩ C ∩ D') = 10 - 1 = 9
n(A ∩ B ∩ C' ∩ D) = 10 - 1 = 9
n(A ∩ B' ∩ C ∩ D) = 10 - 1 = 9
n(A' ∩ B ∩ C ∩ D) = 10 - 1 = 9
n(A ∩ B ∩ C' ∩ D') = 100 - 9 - 9 - 1 = 81
n(A ∩ B' ∩ C ∩ D') = 100 - 9 - 9 - 1 = 81
n(A' ∩ B ∩ C ∩ D') = 100 - 9 - 9 - 1 = 81
n(A' ∩ B ∩ C' ∩ D) = 100 - 9 - 9 - 1 = 81
n(A' ∩ B' ∩ C ∩ D) = 100 - 9 - 9 - 1 = 81
n(A ∩ B' ∩ C' ∩ D) = 100 - 9 - 9 - 1 = 81
n(A ∩ B' ∩ C' ∩ D') = 1000 - 81 - 81 - 81 - 9 - 9 - 9 - 1 = 729
n(A' ∩ B ∩ C' ∩ D') = 1000 - 81 - 81 - 81 - 9 - 9 - 9 - 1 = 729
n(A' ∩ B' ∩ C ∩ D') = 1000 - 81 - 81 - 81 - 9 - 9 - 9 - 1 = 729
n(A' ∩ B' ∩ C' ∩ D) = 1000 - 81 - 81 - 81 - 9 - 9 - 9 - 1 = 729

Thus, all together there are 4(729) + 6(81) + 4(9) + 1 = 2,916 + 486 + 36 + 1 = 3,439 members in the groups.

Therefore, there are all together 3,439 members in the groups.