This question concerns calendar dates of the form \[d_1d_2/m_1m_2/y_1y_2y_3y_4\] in the order day/month/year.

The question specifically concerns those dates which contain no repetitions of a digit. For example, the date \(23/05/1967\) is such a date but \(07/12/1974\) is not such a date as both \(1=m_1=y_1\) and \(7=d_2=y_3\) are repeated digits.

We will use the Gregorian Calendar throughout (this is the calendar system that is standard throughout most of the world; see below.)

  1. Show that there is no date with no repetition of digits in the years from \(2000\) to \(2099\).
  2. What was the last date before today, \(03/11/2010\), with no repetition of digits? Explain your answer.
  3. When will the next such date be? Explain your answer.
  4. How many such dates were there in years from \(1900\) to \(1999\)? Explain your answer.

[The Gregorian Calendar uses 12 months, which have, respectively, \(31\), \(28\) or \(29\), \(31\), \(30, 31, 30, 31, 31, 30, 31, 30\) and \(31\) days. The second month (February) has \(28\) days in years that are not divisible by \(4\), or that are divisible by \(100\) but not \(400\) (such as \(1900\)); it has \(29\) days in the other years (leap years).]