What are the odds of 3 out of 8 people sharing the same birthday?

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I work on a team of 8 people. Three share the same birthday. I'm no mathematician but I imagine the odds are literally astronomical in measure. A figure in this instance may be meaningless to me. Does the answer give a clue to how many times in human history has this likely happened? Could we be the 1st or is this more likely than common sense assumes.

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The chance of a particular three people sharing a particular birthday is $\frac 1{365^3}$. If we ignore the chance four people will share a birthday or two triplets will exist, which will cause overcounting, we can multiply by the ${8 \choose 3}=56$ ways to select three people and $365$ birthdays we could choose, leaving $\frac {56}{365^2}\approx 0.00042$ or $1$ in about $2380$. Not such a low chance at all.