I am having difficulty solving this problem. I am also confused over the meaning

Question

I am having difficulty solving this problem. I am also confused over the meaning of "The sum of the products computed at each step". A clear solution would be of great help. Thanks.

Here is the original problem:

Suppose you begin with a pile of n stones and split this pile into n piles of one stone each by successively splitting
a pile of stones into two smaller piles. Each time you split a pile you multiply the number of stones in each
of the two smaller piles you form, so that if these piles have r and s stones in them, respectively, you compute
rs. Show that no matter how you split the piles, the sum of the products computed at each step equals n(n . 1)/2.

Explanation / Answer

Proof by Induction.

Base case: Suppose we have a pile with 2 stones. There is only one way to split this pile (1<->1)

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