How many greetings can be exchanged between $12$ people if each person only greets each of the others once?

Question

I have this statement:

How many greetings can be exchanged between $12$ people if each person only greets each of the others once?

The correct answer is $66$, but, it really has been difficult for me.

I want to use the combinatorics formula (without repetition): $\frac{n!}{(n - r)!r!} $

But I have, $ n = 12$, but not $r$. How can I get the $r$ and solve this?

Answer 1

The problem is equivalent to find out the numbers of distinc pairs among $12$ that is $\binom{12}{2}=66$.

We can also think about it considering that the first one greets the other $11$, the second one the others $10$ and so on that is $11+10+9+...+2+1=66$.

Answer 2

you need two people to greet each other. we can do this in $\binom{12}{2}$ ways.

Answer 3

It's $2$. The number of greetings is the numbers of distinct pairs, i.e. number of ways you can pick $2$ people out of $12$