How many possible permutations are possible if ranking n entities using the 'standard competition ranking' strategy? I don't know if I'm missing something here, but this doesn't look as straightforward to me as I thought it to be. I basically want to calculate the number of unique rankings that are possible when ranking n entities via the standard competition ranking method. Just to get you started, there can be m distinct ranks that repeat variable number of times, effectively eliminating other possible ranks with each repetition. For reference: Standard competition ranking would rank values like 3, 5, 5, 8 as **1, 2, 2, 4**

I calculated to n=5, and from the pattern, it appears that the number of possible rankings is equal to 2^(n-1).

It's easiest to understand why if you see that for each n, the possibilities can be represented by a binary tree. For example, for n=1, you have a tree with only a single node (value = 1). Each time n increases, you would add another level to the tree, and each parent node will have exactly 2 children: either the same value as the parent or the value of the current level.

So, for n=2, you have 2 children underneath, one with value 1 (same as parent), and one with value 2 (value of the current tree level).

If you continue this process, you will see that each time you increase n by 1, you double the number of possible paths through the tree. Each path through the tree corresponds to a possible ranking. I hope that helps.

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