A path is simple if it repeats no vertices. How many simple paths between two vertices in Complete graph?One way is listing the simple paths is to use depth-first search. but i think it should be simple to find the number of all path between 2 nodes in Complete graph.Here is same problem but mine is for Complete graph:Algorithm that finds the number of simple paths from $s$ to $t$ in $G$
Count of all simple paths between two vertices in a Complete graph
graphs;combinatorics
$c(1)=0,$$c(n)=(n-2)*c(n-1)+1$suppose we have c(n-1) path in (n-1)-Complete Graph.when we add one more vertices, we add $(n-2)*c(n-1)$ new path and one for directed path.the result is same as real:01251665===============================EDITED$c(n)=\sum_{i=0}^{n-2} \frac{(n-2)!}{i!}=(n-2)!*\sum_{i=0}^{n-2}\frac{1}{i!}$ .This shows that's about $(2\text{}3)*(n-2)!$