Question 23
Let $p$ be a prime number and $n$ a positive integer such that $n \ge p$. Show that $\binom{n}{p} - \lfloor \frac{n}{p} \rfloor$ is divisible by $p$. (Here the combinatorial symbol $\binom{n}{p} = \frac{n!}{p!(n-p)!}$ and $\lfloor x \rfloor$ stands for the largest integer less than or equal to $x$.)