Number of pairs containing a unique number Say you have $n$ different numbers/balls/something and you choose 2. i.e. you have $\binom{n}{2}$ different possibilities to choose. Now my question is how many of those contain a certain number/ball/something?
e.g. say you have $n=6$: $\binom{6}{2} = 15$. There are exactly 5 pairs that contain a certain number $i$. And in general the answer is $n-1$. I found this using induction. The question is if there's a better and more intuitive reason why is that so.  

After you fix the first item, you have $n-1$ possibilities for the second item because you can't select the same item twice.
