1. (10 points) We will call a date diverse if in its representation in the form DD/MM/YY (day-month-year) all digits from 0 to 5 are present. How many diverse dates are there in the year 2013?
Answer: 2.

Solution. Note that in any date of 2013 in the specified format, the digits 1 and 3 are used, so for the day and month of a diverse date, the digits left are 0, 2, 4, and 5.

Let $Д_{1}$ and $Д_{2}$ be the first and second digits in the day's notation, and $\mathrm{M}_{1}$ and $\mathrm{M}_{2}$ be the first and second digits in the month's notation. Since there are 12 months in a year, $\mathrm{M}_{1} \leqslant 1$, hence $\mathrm{M}_{1}=0$. Moreover, no month has more than 31 days, and of the remaining digits, only 2 is suitable for $Д_{1}$. The digits 4 and 5 can be placed in positions $Д_{2}$ and $\mathrm{M}_{2}$ in any order, giving a valid date.