This is a classic . . . The Birthday Paradox.
. . I'll do the easy one.
Thirty people write down the day and month of their birth.
a) What is the probability that 2 or more of these dates will be the same ?
The opposite of "at least one match" is "no matches."
Person #1 can have any birthday: .
#2 can have any of the remaining 364 days: .
#3 can have any of the remaining 363 days: .
. . . and so on . . .
#30 can have any of the remaining 336 days: .
. . (I'll let you crank it out.)