Birthday paradox with a twist
I understand the classic birthday paradox problem, but how does the math change if I have a room of 34 people and I want to know the probability that five of them share the same exact birthday, like all 5 are born on June 15th? This is different in that I'm not asking the probability that 5 people with different birthdays each find a match in the room.
I'm not sure how to formulate the probability of the 5 June 15th birthdays NOT happening and then subtracting from 1.
Thank you!
Edit: sorry, the date is arbitrary. The odds of 5 people sharing ANY date is what I'm after.