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 = {}` ?

Answer: =58968

Why:

Each of the numbers 1,2,3,....10 belongs to one of the sets- we assume no empty set due to the teachers demand.

Let ABBBCACCCB denote situation where 1 and 6 are in A , 2 3 4 AND 10 in B etc

Clearly it isomorphic to a base- THREE 10 digit presentation where all 3 digits have to be present

all possibilities: 3^10

two digits only 3* (2^10-2)

one digit only 3

Answer: 3^10-3* 2^10 - 6-3 =59049- 3*1022 - 3=58968

ady

*Edited on ***February 10, 2004, 10:44 am**