Puzzle #113: Date Digits

Here are two puzzles dealing with dates.

If a date is expressed as digits, there are a number of formats that can be used. One, called ISO 8601, is YYYY-MM-DD. For some years, there are a number of dates where all eight digits are different. For 2013, however, there were no such dates. Why not?

If the format YY-MM-DD is used instead, there were dates in 2013 where all six digits were different. How many such dates were there?

Submit your answer to Gene Wirchenko <genew@telus.net>. Your answer should be in the form of a proof. That means to show how your answer must be correct. The deadline is Wednesday, August 5, 2015 at noon Pacific Time. I will post the answer shortly after.