Dates with 8 consecutive digits In many places, dates are written as DD/MM/YYYY. For example, the 25th of April 1736 is written as 25/04/1736. Dates such as this one that use 8 consecutive digits (not necessarily in order) will be called illions. 


*

*What is the first illion after 2015?

*Why must there be a 0 in every illion in the years 2000 to 2999?

*Why must every illion in the years 2000 to 2999 have 0 as the first digit of the month?

*How many illions are in the years 2000 to 2999?


(You cannot use the same digit twice in an illion)
I have tried to list all of the possibilities however, I have found this highly time consuming. I was wondering what method I should use to solve these questions without listing all the possibilities.
Thank you :)

(1) As argued in (2) below, the month contains a $0$ or is $12$, hence either the year is $>2999$ or does not contain a $0$. This makes $2134$ the earliest possible year. However, the leading digit of the day is $0$, $1$, $2$, or $3$, so that we must up the year to $2145$ at least, which allows day $30$ but then conflicts with the $0$ in the month. The next option is that the year does not use $1$ (nor $0$), which happens for the first time in $2345$. We can find an illion in that  year: We know we use $0$ in the month (preferably as leading digit). Then the leading digit of the day must be $1$. Now the lowest valid month is $06$ and we can take $17/06/2345$
(2) If the two-digit month does not contain a zero and has two distinct digits, it must be December. But then the use of $2$ conflicts with the leading $2$ of the year. Actually, we see that the same argument applies to the year range $1000$ to $2999$.
As in (2), we know that the month cannot be $12$ or $11$. In cannot be $10$ either, because that makes $0,1,2$ used up by month and year, hence the day must have leading digit $3$, but neither $30$ nor $31$ are allowed
For (4), you may want to first enumerate valid day/month combinations (without $2$!) and look for years that fit the gaps.
