ЕЕооокимов M.A. Anya calls a date beautiful if all 6 digits in its notation are different. For example, 19.04.23 is a beautiful date, while 19.02.23 and 01.06.23 are not. How many beautiful dates are there in 2023?
The digits 2 and 3 are already part of the year number, so among all the months, only $01, 04, 05, 06, 07$, 08, 09, and 10 need to be considered. Each of these month numbers contains a 0, so in a beautiful date, there will be no day number starting with 0, 2, or 3, and there will also be no days 10, 11, 12, and 13—leaving only 6 days, from 14 to 19. However, then in each month, a beautiful date starts with 1, and only 6 months, from 04 to 09, are suitable. It remains to note that for each suitable month, exactly one day ending in the same digit will not be beautiful—meaning, in each of the 6 months, there are 5 beautiful dates, and in total in 2023—30.

## Answer

30.