Eccoci.

A set of disks numbered $1$ through $100$ are placed in a line in random order.

What is the probability that we have a partial derangement such that exactly $22$ prime number discs are found away from their natural positions?
(Any number of non-prime disks may also be found in or out of their natural positions.)

Give your answer rounded to $12$ places behind the decimal point in the form 0.abcdefghijkl.