To demonstrate set union and intersection to her class, Mrs. Putnam asked for three students to each write down a set of numbers.
After they had done so, she looked at their sets and told the class, "the union of these three sets is the first ten counting numbers, but their intersection is empty!"
How many triples (A, B, C) of sets are there such that
A U B U C = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}
and
A ∩ B ∩ C = {} ?
Since there are three sets, a number can be in two of them and still not be in the intersection of all three. Then, for any one interger, there are six different ways it can be found among A, B, and C: only in A, only in B, only in C, in A and B, in A and C, in B and C.
In total there are 6^10 =
60466176 possible triplets (A, B, and C).
This number includes empty sets. There are 3^10 triplets with A empty, likewise for B and C. The set with A, B empty was counted twice, likewise with A, C and B, C.
The total number of triplets with empty sets is 3*3^10-3 = 177144
The total number of triplets with no empty sets is 60466176 - 177144 =
60289032.
Solution
