This is a repost of this math.se question that I am posting here since it received no attention there.

What is the probability that a random edge coloring of $K_n$ with $m \geq n$ colors results in a proper coloring ?

By random edge coloring I mean that every edge is assigned a color from $\{1,\ldots,m\}$ uniformly at random.

If we define the event $E_i$ to mean that edge $e_i$ is not incident with an edge of the same color class of the event $V_i$ that vertex $v_i$ is incident with edges of distinct colors, then it is easy to get a bound for the above probabiity in terms of $Pr[E_i]$ or $Pr[V_i]$ by using the union bound.

The problem is that it appears that such approach only gives bounds that make sense only when $m = \mathcal(n^2)$ which is not interesting.

Hence I am wondering if there is a more sensitive way to bound/compute the above probability?