How can we calculate the number of ways in which a $p$-sided polygon can be made inside a $n$-sided polygon such that no side of the $p$-sided polygon is in common with the $n$-sided polygon?
2026-03-25 21:53:29.1774475609
Permutation polygon
128 Views Asked by Bumbble Comm https://math.techqa.club/user/bumbble-comm/detail At
1
Pair each one of the $p$ vertices with the adjacent vertex in a clockwise direction. Then all we have to do is count the number of ways of selecting $p$ pairs of adjacent points from the $n$ available.
Let $v$ be one of the $n$ vertices.
Case 1 When $v$ is not one of the $p$ vertices we have to choose $p$ pairs from a line of $n$ points. We can think of this as the number of arrangements of $p$ pairs and $n-2p$ single points in a line.
Case 2 When $v$ is one of the $p$ vertices we have to choose a further $p-1$ pairs from a line of $n-2$ points.
The total is $\begin{pmatrix}n-p-1\\p-1\\\end{pmatrix}+\begin{pmatrix}n-p\\p\\\end{pmatrix}$.