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(3): Solution FINAL WORD(by Charlie by Ady TZIDON)

My formula is f(n) = 6^n - (3*3^n - 3) if no empty sets are allowed.

f(1) = 0 is correct since at least one of A, B and C must be empty for the intersection to be empty.

f(2) = 36 - (3*9 - 3) = 12. The triples (A,B,C) are:

({1},{1},{2})

({1},{2},{1})

({2},{1},{1})

({1},{2},{2})

({2},{1},{2})

({2},{2},{1})

({1,2},{1},{2})

({1,2},{2},{1})

({1},{1,2},{2})

({2},{1,2},{1})

({1},{2},{1,2})

({2},{1},{1,2})

f(3) = 138 and f(4) = 1056 are longer to list.

f(3) = 216 - (3*27 - 3) = 138

f(4) = 1296 - (3*81 - 3) = 1056