Score : $300$ points
There are $N$ towns in a coordinate plane. Town $i$ is located at coordinates ($x_i$, $y_i$). The distance between Town $i$ and Town $j$ is $\sqrt{\left(x_i-x_j\right)^2+\left(y_i-y_j\right)^2}$.
There are $N!$ possible paths to visit all of these towns once. Let the length of a path be the distance covered when we start at the first town in the path, visit the second, third, $\dots$, towns, and arrive at the last town (assume that we travel in a straight line from a town to another). Compute the average length of these $N!$ paths.
Input is given from Standard Input in the following format:
$N$ $x_1$ $y_1$ $:$ $x_N$ $y_N$
Print the average length of the paths. Your output will be judges as correct when the absolute difference from the judge's output is at most $10^{-6}$.
3 0 0 1 0 0 1
2.2761423749
There are six paths to visit the towns: $1$ → $2$ → $3$, $1$ → $3$ → $2$, $2$ → $1$ → $3$, $2$ → $3$ → $1$, $3$ → $1$ → $2$, and $3$ → $2$ → $1$.
The length of the path $1$ → $2$ → $3$ is $\sqrt{\left(0-1\right)^2+\left(0-0\right)^2} + \sqrt{\left(1-0\right)^2+\left(0-1\right)^2} = 1+\sqrt{2}$.
By calculating the lengths of the other paths in this way, we see that the average length of all routes is:
$\frac{\left(1+\sqrt{2}\right)+\left(1+\sqrt{2}\right)+\left(2\right)+\left(1+\sqrt{2}\right)+\left(2\right)+\left(1+\sqrt{2}\right)}{6} = 2.276142...$
2 -879 981 -866 890
91.9238815543
There are two paths to visit the towns: $1$ → $2$ and $2$ → $1$. These paths have the same length.
8 -406 10 512 859 494 362 -955 -475 128 553 -986 -885 763 77 449 310
7641.9817824387