# birthday problem (Q339000)

mathematical problem
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English
birthday problem
mathematical problem

## Statements

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{\displaystyle {\begin{aligned}{\bar {p}}(n)&=1\times \left(1-{\frac {1}{365}}\right)\times \left(1-{\frac {2}{365}}\right)\times \cdots \times \left(1-{\frac {n-1}{365}}\right)\\&={365\times 364\times \cdots \times (365-n+1) \over 365^{n}}\\&={365! \over 365^{n}(365-n)!}={\frac {n!\cdot {365 \choose n}}{365^{n}}}={\frac {_{365}P_{n}}{365^{n}}}\end{aligned}}}
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