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

Game, set and match !!!! by Penny)

Penny,

You wrote:

"*If C has just 1 number (there are 10 such sets), then any two sets A and B, neither of which have that number, will work.
*

10*(3^9) = 196830 "

Though that is true, any two sets, where **not both** have that number will also work.

So, if C = {1}

Then as long as *at least* one of set A and B don't have it (and they include all the remaining 9 numbers), it will ALSO produce a valid solution.

(e.g., A = {1, 2, 3, 4, 5, 6, 7} and B = {2, 8, 9, 10} )