Derangement is a permutation without a fixed point.

Derangement - Wikipedia, the free encyclopedia
The probability that $\displaystyle n$ elements are deranged is given by $\displaystyle \sum_{k=2}^n \frac{(-1)^k}{k!}$. So the probability exhibits the following properties:

(1) When $\displaystyle n$ 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 $\displaystyle n$ 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 $\displaystyle n$ is restricted to odd numbers, the probability is an increasing function of $\displaystyle n$.

(4) When $\displaystyle n$ is restricted to even numbers, the probability is a decreasing function of $\displaystyle n$.

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