Question

Given 4 points on a circle, how many ways can two points be connected, using permutations / combinations to solve?

This does not bring up anything useful on google, making it even more difficult for me to solve.

Answer 1

`3! = 6`

Explanation:

Let's first approach this by naming the points - I'll use the compass directions North, South, East, West.

Let's notice that NS and SN are one and the same connection and so we only want to count that once.

Therefore, starting from N, we have NE, NS, and NS.

We can move the starting point to E and have ES and EW (EN has already been done)

And then we have SW.

A total of 6 connections can be made.

We can get to this result a different way. This problem is akin to handshakes - if NE shakes hands, and then NS, and then NW, that's 3 handshakes. E then starts to shake hands but we don't want to repeat the shake to N, so the unique shakes are ES and EW, which is 2 handshakes. And then SW does the last unique handshake. That's

`3! = 6`

And so if we had a substantially larger table with many more circles/seats, we can generalize the number of unique connections/handshakes to be:

`("Circles/seats/things in a circle"-1)!`