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 = {}` ?

(In reply to

re(2): Game, set and match !!!! by Penny)

"*If C has just 1 number (there are 10 such sets), then any two sets A and B, where A is any subset of the remaining 9 numbers, and B is any subset of all 10 numbers, should do the trick. A set of 9 elements has 2^9 subsets. A set of 10 elements has 2^10 subsets.
*

(10)*(2^9)*(2^10)=5242880"

and for completeness...

where B is any subset of the remaining 9 numbers and A is any subset of all 10 numbers, should also do the trick.... (I just switched A and B in what Penny said--quoted above.)