How many line segments are drawn?

Question

Twenty-four points are placed around a circle. A line segment is drawn between each pair of points.

Answer 1

Let’s start with point 1. It can be joined to points 2 to 24, that’s 23 segments.

Now point 2. We’ve already included 2-1, so we have 2-3, 2-4, etc., 22 segments.

Then point 3, 21 segments.

Finally, 23-24, 1 segment.

So the total number is 23+22+21+...+1=(23+1)+(22+2)+(21+3)+...+(13+11)=11×24=264.

To this we need to add 12 for the segments 12-13, 12-14, ..., 12-24, 12 segments, making 264+12=276.

This is the same as using the formula ½n(n+1) where n=23.