Birthday Problem

• $x$ is the number of possible patterns
  • 365 for birthday
• $k$ is the number of patterns

The probability of the collision in the $k$ patterns is

\[\begin{eqnarray} 1 - \frac{ \frac{ x! }{ (x - k)! } }{ x^{k} } & = & 1 - \frac{ x! }{ x^{k} (x - k)! } \nonumber \\ & = & \frac{ x^{k} (x - k)! - x! }{ x^{k} (x - k)! } \nonumber \end{eqnarray}\]

Reference

• https://en.wikipedia.org/wiki/Birthday_problem