Twenty-two Foolish Primes

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.

Deranging a set means permuting the set such that none of the elements remain in their original place. To solve this problem, we need to consider the derangements of the prime number disks specifically, and then consider the fact that all the other disks can be arranged in any order.

We start by determining how many prime numbers there are in the first $100$ natural numbers. There are exactly $25$ prime numbers in this set: $2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89,$ and $97$.

The problem statement asks for the exact $22$ prime number disks to be out of position. So, we look for ways to derange these $22$ primes in $25$ locations.

Using the inclusion-exclusion principle for derangements, the number of derangements of $n$ items is given by

$D(n) = n! \left( \frac{1}{0!} – \frac{1}{1!} + \frac{1}{2!} – \frac{1}{3!} + … + (-1)^n \frac{1}{n!} \right)$

We can calculate $D(22)$ using this formula. We also need to calculate the number of ways to choose 22 primes out of 25. This is found using the binomial coefficient, $\binom{25}{22}$.

Therefore, the number of ways $22$ prime number disks can be deranged is $\binom{25}{22} * D(22)$.

The total number of arrangements of the $100$ disks is simply $100!$.

So, the probability is given by

$p = \frac{\binom{25}{22} * D(22)}{100!}$.

When it comes to calculating that number, it is unfortunately beyond human capabilities to do it without the use of a computer or calculator. Nevertheless, you can use a software program or calculator to get the required precision of $12$ decimal places.

Hence, to solve this problem, make use of the available software or calculators that can handle large factorials and precise calculations.

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