This is a "pile-splitting problem". Here is the situation: you start with n objects in a pile, and split it into two smaller piles. Then, for each new pile, you continue the splitting process until (finally) there are n piles of size one. At each splitting operation, you compute the product of the size of the two smaller piles. Once there are n piles, sum up all the products computed. The result will be a function of n, and will not (as it turns out) depend on how each pile is split along the way. Conjecture what this function is, and how to prove that our conjecture is correct.
Here's an example, for n = 5.
Split the pile into 3 and 2. (The product of this split is 6.)
Split the 3 pile into 2 and 1. (The product is 2.)
Split the first 2-pile into 1 and 1. (The product is 1.)
Split the second 2-pile into 1 and 1. (The product is 1.)
Now, the total is 6 + 2 + 1 + 1 = 10
Suppose you have a pile of n objects, each two of them are tied with a rope. You divide the pile into two piles (having m and k objects) and cut all the ropes which link objects from different piles. Number of the ropes you cut at this operation is m * k.
Now you continue the process until each pile has exactly 1 element. All the ropes would be cut by that time. And this is exactly the sum of all the products computed in the process. And this is
n * (n - 1 ) / 2.