How many even three-digit positive integers have the property that exactly two of the integer's digits are 6's?
Suppose the first two digits are 6's.
The only possibilities are 660, 661, ..., 666, 667, ..., 669.
You can count the other possibilities by hand. Just make sure that you don't count 666 three times and 0 cannot be the first digit, and you're good to go.
Although 661, 667, 669 are possibilities with regard to the 2-six requirement, they can't be
part of the solution because the problem says "How many even three-digit positive integers"