### 46th Putnam 1985

**Problem A1**

How many triples (A, B, C) are there of sets with union A ∪ B ∪ C = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} and A ∩ B ∩ C = ∅?

**Solution**

Answer: 6^{10}.

*Easy.*

Careful. The answer is not 3^{10} (this trap is the only reason I do not rate the problem trivial!).

There are six possibilities for each element: just A, just B, just C, not A (meaning B and C, but not A), not B, or not C.

46th Putnam 1985

© John Scholes

jscholes@kalva.demon.co.uk

3 Nov 1999