How many Z+ ≤ 1000 have distinct digits and are even?

Solution.

Single digit numbers:        _____

                                    {2, 4, 6, 8} = 4 digits

Double digit numbers:       _____   _____

                                    4 digits  {0, 2, 4, 6, 8}

                                                5-1=4 digits

                                                exclude the digit used in the first position

Triple digit numbers:        _____   _____   _____

                                    4 digits 4 digits 5-2=3 digits

                                                           exclude the 2 digits already used in previous positions

The only four digit number we have is 1000, but it does not meet our conditions.

So we have, 4 + (4 x 4) + (4 x 4 x 3) = 68 distinct even digits ≤ 1000.

Don't you feel like this sometimes when you write proofs?! Ha, writing proofs can seem nutty <:-)