medium or not?

Hi everybody,

I am trying to solve a mathematical problem that came out of a TV programme I was watching; I will explain how this happened, and I hope someone can help (I believe the problem has long been solved).

I think the programme was called 'psychic challenge', a sort of game where they wanted to check whether some people were 'real' mediums or not by putting them through some tests.

One of the games consisted in showing the 'medium' 5 men and 5 women, and knowing that they were 5 couples, match each wife to her husband.

One of the mediums guessed just one out of five, and everybody was happy with that, but then it occurred to me that if you matched these people at random, guessing only one couple right would be more probable than guessing all of them wrong, i.e. you would be more of a 'medium' if you didn't guess any of them right!

Then I tried to write a formula to calculate the probability (I'm not a mathematician, so please excuse any blunder).

I will not go through all the details of my reasoning; I will just say that first I realised that the usual conditional probability formula didn't work here, so I went back to basics (set theory), expressed the probability of a sequence of dependent events in terms of independent ones, and ended up with this formula:

[math]P(k,N) = \frac{1}{k!} \sum_{i=k}^{N}\frac{(-1)^{i-k}}{(i-k)!}[/math]

This would be the probability that, matching at random the N wives to the N husbands one gets k (and only k) of them right. Applying it to N = 5 yields:

[P(0,5) = 11/30, P(1,5) = 3/8, P(2,5) = 1/6, P(3,5) = 1/12, P(4,5) = 0, P(5,5) = 1/120]

which is indeed correct if you enumerate the 120 possible dispositions and count them.

My question is: is there a simplified version of the formula, like a closed form of the alternating factorial summation?