Induction principle: container with $n$ objects

Question

Given a container of $n$ objects, prove for induction that for every integer $n≥1$, the number of ways you can choose 2 objects between the $n$ objects of the container is $\frac{n(n-1)}{2}$.

For $n=1$ I get $0$ but this doesn't prove the inductive hypothesis. How can I prove it?

Answer 1

With $n+1$ items, designate some specific item. There are $n$ pairs that contain this item. By the induction hypothesis, there are $\frac{n(n-1)}{2}$ pairs that do not contain the item. Add them up to verify the formula for $n+1$.

Answer 2

The basic induction process works like this.

1. Test base case (which you did for n = 1).
2. Assume the hypothesis: Given $n$ objects, the number of ways is $\frac{n(n-1)}{2}$.
3. Show that the result holds for $n+1$ objects. You need to show that the number of ways is $\frac{(n+1)n}{2}$.