I intend a simple path to be a path in a graph in which an element does not appear twice.
Now, let $ G = \langle V,E\rangle $ be a graph.
My professor told us that the maximum number of simple paths in a graph is $$\sum_{i}^{|V|}(i!)$$ Where $i$ denotes the length of the path.
I would say it's not like this. If $|V| = 1000$ and $i=3$ seems I can create only 6 simple paths when the length of it is 3, which I don't think it's the case.
I think the exact formula would be $$\sum_{i=0}^{|V|}\frac{|V|!}{(|V|-i)!}$$ Which sums the number of k-permutations of n for each length of the path. Am I wrong? Thank you.