Having some trouble with this combinatorics problem

[Calabi_Yau]

6 friends go to a party, each one carrying a different umbrella. They place the umbrellas outside. When the party is over, they are drunk and each one grabs an umbrella at random.
In how many ways could none of them have taken the right umbrella?

I'm having a bit trouble with this, as I can't seem to solve it without having to do some rough counting some times. Can any of you bother to solve this and explain it to me?

 

[Robert1986]

Take a look at this link: http://www.proofwiki.org/wiki/Hat-Check_Problem

 

[Norwegian]

You are asking about the number of derangments S_n of a set with n elements. If you are looking for an exact answer, you can either use the recurrence relation S_n+1=n(S_n+S_n-1), or compute the alternating sum Ʃ(-1)ⁱn!/i! where i goes from 0 to n.

 

[Calabi_Yau]

Thanks guys.