Divisibility of binomial coefficient

Question

Let $p$ be a prime number and $q\in \{1,\dots,p-1\}$. Prove that $\tbinom{2p-q-1}{p-q} \equiv 0\pmod {p}$

However, I have no idea how to prove this.

Would be thankful for solution.

Answer 1

Assuming $C^n_k=\binom{n}{k}$:

$2p-q-1>p-q$, so you just have $C^{p-q}_{2p-q-1}=0$ $\dots$

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Also $C_{p-q}^{2p-q-1}=\frac{(2p-q-1)!}{(p-q)!(p-1)!}$ is divisible by $p$, since $p-q,\ p-1\in\{0,\dots,p-1\}$ and $2p-q-1\ge p$.