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 = {} ?

For ordered triples:
Each of the elements belongs to one of the three sets; there are 3^10=59,049 ways of doing this.

For unordered triples:
Most of the triples appear 3!=6 times as ordered triples. One, ({1, 2, 3, 4, 5, 6, 7, 8, 9, 10},{},{} ), appears only three times, so before dividing by 6 we have to add 3 in order to treat this one case as if it appeared 6 times.

(59049+3)/6 = 9842 unordered triplets.
Edited on February 10, 2004, 10:46 am

Posted by Charlie on 2004-02-10 10:45:08