# co.combinatorics – Probability permutation in turned to cycle

If $$M$$ is $$ntimes n$$, the the probability is $$1/n$$.

Indeed, by symmetry, the probability in question is the probability that a random permutation of $${1,dots,n}$$ is a complete cycle. There are $$n!$$ permutations and $$(n-1)!$$ complete cycles. So, the probability is $$(n-1)!/n!=1/n$$.