combinatorics - Proof through combinatorial argument

I am attempting to solve this counting problem through combinatorial argument. The following is the equation I am given:

$$\sum_{i=1}^n (i-1)(n-i) = \binom{n}{3}$$

I understand that the right-hand side of this equation represents a set of $n$-elements out of which we choose 3. For example I believe we can say suppose we have a group of $n$ people and we want to choose 3 out of $n$ to be in a committee. However I'm not sure how to express the left-hand side in words. If forming a committee is an appropriate way to tackle this problem then I know the left-hand side must utilize the addition and multiplication principles, but I don't know how to put it into words. Also my intuition tells me that in solving this we should first flip $$(n-i)(i-1)$$

Thanks!

Answer

Hint: split on the fact that the middle member (in sorted numerical order, the members are numbered $1$ to $n$) is $i$. Then we pick one from before and one from after.