a set A of n elements has n(n-1)/2 subsets of 2 elements

dpesios

I would very much like some help to the following problem.

1. The problem statement, all variables and given/known data

Using mathematical induction, prove that a finite set A of n elements has n(n-1)/2 subsets of two elements.

3. The attempt at a solution

* Base step n=2: 2(2-1)/2= 1 subset of two elements.
* Inductive step: assuming the statement holds for n=k, that is a set A of k elements has k(k-1)/2 (hypothesis)
We want to show that it also holds for n=k+1, that is a set A of k+1 has (k+1)(k+1-1)/2 elements.
How can we infer from the hypothesis ??? I have no idea ...

I have an engineering background so be as descriptive as you can.
Thank you in advance.

eumyang

A set A of k elements has k(k-1)/2 subsets of two elements, as you said.
Suppose you add a new element to set A. How many new subsets can be created where one of the elements of these subsets is the new element?

dpesios

if we add an element to the set which previously had k elements (that is now has k+1 elements) the new subsets that include the new element will be :

(k+1)k/2 - k(k-1)/2 = k

So, how can we argue that this will solve the problem ?