Derangement is a permutation without a fixed point.

Derangement - Wikipedia, the free encyclopedia
The probability that

elements are deranged is given by

. So the probability exhibits the following properties:

(1) When

increases from an odd number to an even number (say, from 3 to 4), the probability increases. That is, when elements are shuffled, they get more likely to be deranged.

(2) When

increases from an even number to an odd number (say, from 4 to 5), the probability decreases. That is, when elements are shuffled, they get more unlikely to be deranged.

(3) When

is restricted to odd numbers, the probability is an increasing function of

.

(4) When

is restricted to even numbers, the probability is a decreasing function of

.

I want, if any, "intuitive" explanations of these properties.