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How do I calculate the odds that, for X number of people, every possible birthday is accounted for?

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How do I calculate the odds that, for X number of people, every possible birthday is accounted for?
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This is a famous problem in statistics called the Coupon Collector’s problem. For what it’s worth, the most likely number of people needed in order for all 365 birthdays to be represented is 2365.

So, surely with 10, 20, 30 times that many people it would be quite likely.
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First, think about a specific day. What is the chance that no one out of n people was born that day? That is, they were all born on some other day.

Then, what is the chance that at least one person of n was born that day?

Finally, with which operation do you say that at least one person was born on the 1st day of the year OR the 2nd OR ... OR the 365th.
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