Question 90
Let $n\ge4$ and $k,d\ge2$ be natural numbers with $kd\le n$. The $n$ participants in the Mathematical Olympiad sit around a round table waiting for Patrick. When Patrick arrives, he is unhappy because the social-distancing rules are not being followed. He chooses $k$ of the $n$ participants to stay and sends all the others out of the room, so that between any two consecutive remaining participants there are at least $d-1$ empty seats. How many ways can Patrick do this, assuming that all seats were initially occupied?