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.

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u/DonutHoleTechnician — 2 days ago

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u/DonutHoleTechnician — 26 days ago

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