Rencontres numbers

In combinatorics is meant by Rencontres Number (French encounters ) the distinguishable with designated number of permutations of a lot of elements in the meticulously maintained elements of their original place or purely coincidental " find ":

In the event that none of the elements retains its place and " finds ", arises as a special case the Subfakultät, a formula for the number of possible fixed-point- free permutations (also derangements or "Total dislocations " ) of the elements, ie in which none of them at his previous place remains:

Example

A car owner has the engine of his new four- cylinder cleaned and forgot to write down that on which spark plug is one of the four spark plug wire. How many possibilities are there for exactly two of the four cables properly aufzustecken?

In detail:.

A year later, it happened the same with the engine of his new six- cylinder. How many possibilities are there now, reattach it exactly half of the spark plug wire right?