I cannot find a reference for the following fact: $${n - 1 \choose p - 1} \equiv \begin{cases} 1 \;\mbox{(mod }p\mbox{ )} & \text{if } \; p \vert n \\ 0 \;\mbox{(mod }p\mbox{ )} & \text{if } \; p \nmid n. \\ \end{cases}$$ where $n$ is a positive integer and $p$ is a prime number. Could you suggest any book or paper in which such a property is stated and proved?
Thanks in advance for any help on this.
Best wishes.