## Permutations

Hello every one,

I've been given the following task:

We consider 100 people each with a given number from 1 - 100. Then on 100 notes there are written one number, each note with a distinct number also from 1 - 100. These notes are then shuffled and put into 100 different boxes. With each a number on it (also 1 - 100).

Then each person is to go and find their own identifying number in the boxes within fifty tries. If a person succeeds then they move on, with no communication possibilities with the remaning people, also if only one of the 100 people fail, it counts as a failure for everyone.

The question is then:
If each person pick their own identifying number as the first box, then if the number of the note in the box is theirs good they succeed, else they pick the next box matching the note within the first and so on. What is the chance og succes for all the people?

It should be mentioned that the initial configuration of the numbers are not interchanged from person to person.

So far I've figured out that the initial configuration correspond to a permutation in $S_{100}$. And that any permutation corresponding to a cycle of length 50 or less means succes whereas anything greater than fifty means failure for everyone.

Then i try to calculate all the possible cycles leading to success as:

$\sum \limits_{k=1}^{50} \frac{n!}{(n-k)!k}$

Where $n$ is given by the permutation group $S_{100}$, and $k$ denotes the length of a cycle. Then having obtained the above number I divide by $100!$ because that is the total number of permutations in $S_{100}$, but then the chance of success is only approximately $6.7 \cdot 10^{-67}$. Which seems like an awfully small success rate? Can any one confirm my result or tell me what I am doing wrong?