It's my birthday, and I figured I will create a problem based on birthdays that I myself am unable to solve!
Assuming time is denoted by HH:MM:SS, MM/DD/YYYY, what is the probability that in a class of 191 students, 47 of them have birthdays:
(1) within one week of each other
(2) with the product of the digits in the HH:MM:SS representation equal to the sum of the digits in the MM/DD/YYYY representation
assuming that the earliest year is 1900 and leap years are allowed?
Very low The usual birthday paradox comes up because there are so many pairs of people to have the same birthday. When you want so many to have these matches it becomes difficult. For $2$, you should be able to write a program to find the probability that two will match. It will already be low as a single zero in the time will make sure there is no match. For 1, the chance that $47$ have a birthday in the same calendar week (not quite the same as what you ask) is about ${191 \choose 47}52^{-46}$ which Alphaevaluates as about $1.5 \cdot 10^{-34}$