Help needed urgently!

[reddog843]

This problem is supposed to be proved through induction.
Let P1, P2,...,Pn be n points in a plane with no three points collinear. Show that the number of line segments joining all pairs of points is (n^2 - n)/2.
I am totally lost and would appreciate any help. Thanks!

[Toptunov]

I’m not a mathematician, but we can easily reach the solution by ordinary logic.

The total amount of points is n.

From each point start (n-1) lines (implicit assumption: only open lines; explicit assumption: no overlapping lines).

From all points start n(n-1) lines.

Each line connects “two” vertexes, so the above total is overesteemed the double of its real value:
n(n-1)=2x

x=n(n-1)/2 the solution you were looking for.

[HallsofIvy]

Toptunov's solution is completely correct. This is also sometimes given as the "handshake" problem: if every person in a group of n people shakes hands with each other person, how many handshakes are there? or the "tournament" problem: if each of n teams in a round-robin tournament must play every other team once, how many games are there?
Each person shakes hands with n-1 other people so the n people have n(n-1) handshakes- except that two people share one handshake: number of handshakes,n(n-1)/2.

If you MUST use induction, then, first, check n= 1. If there is only one point there are NO pairs so NO lines: (1^2-1)/2= 0.

Now assume that for some number k, k points means (k^2- k)/2 lines. Add one more point to get k+1 points. Now we must add a line from that new point to each of the k original points: k new lines means that the total will be
(k^2- k)/2+ k= (k^2- k)/2+ 2k/2= (k^2+ k)/2= (k^2+ 2k+ 1-k-1)/2= ((k+1)^2- (k+1))/2 exactly what we need.

[Toptunov]

Thank you HallsofIvy, also for your further explanation.
You seem pretty skilled in mathematics and generous enough to help many students, go on this way!