100 prisoners 100 boxes problem I have some issues with a problem I found on this page: http://www.mast.queensu.ca/~peter/inprocess/prisoners.pdf 
The problem goes as follows:
"A group of 100 condemned prisoners are offered the chance to play the
following game. They will be each be allowed to enter a room one at a
time in any order they wish until all 100 have gone in once. In the
room there is a long table with a row of 100 identical wooden boxes
numbered in order from 1 to 100. Each box can be opened and inside it
there is the name of one of the prisoners, such that each of the 100
names appears exactly once. However, the ordering of the names in the
boxes is random. Once inside the room, each prisoner is allowed to open and look inside
50 boxes—any 50 that he wants. After he has opened his 50 boxes he
must leave the room, making sure that the boxes are in exactly the same
state as he found them, and he is no longer able to communicate with
any other prisoner. The reprieve that they have been granted is that at
the end of the game they will all be spared on condition that every prisoner
manages to find the box that contains his own name. On the other
hand, if at least one of the prisoners fails to find his name, they will all
be executed at dawn. Let’s be clear about this—in order to escape execution,
it is necessary that every prisoner find his own name."
Now there is a strategy so that to probability of success is about 30%. Prisoner 1 goes in the room, opens box 1 and if he finds his number, he can leave, if not, then he opens the box whose number he has drawn in the previous box. He needs to find his number within 50 tries. Then prisoner 2 does the same with box 2 etc.   
My problem basically is the calculation of the number of permutations having a cycle of length k. On page 3 it is said that there are $\binom {100}{k}$ ways to choose the elements of that cycle, $(k-1)!$ ways to arrange the elements and $(100-k)!$ for the remaining elements. I see why it is $(k-1)!$ arrangements  (because it's a cycle and thus there are $(k-1)$ other arrangements that describe the same cycle), but why do we not do this to the remaining elements not being in the long cycle? Why don't we also take every possible permutation of the remaining numbers? Why is it not $(100-k-1)!$?

In the long cycle, we have to make sure it is a single cycle.  For the rest, you do not need to make sure it is a single cycle, so you can permute all $100-k$ elements in any order at all.  Even the identity permutation is acceptable. For the long cycle, you have to make sure you get a single cycle, which is why you only have $(k-1)!$ for $k$ elements
