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

Solution by Brian Smith)

BS's solutiom is correct except for small amendment which I am going to suggest:

He wrote:

..."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..."

My remark : add 3 for the number of "choices" deducted twice- when counting "triplets with no empty sets"

the AAAAA.. choice is deducted when counting A*B & A*C

etc

so the correct answer is 60289032+3= 60289035

60289035

A little, but important correction...

ady