Having some trouble with this combinatorics problem

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    Combinatorics
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Discussion Overview

The discussion revolves around a combinatorics problem involving six friends who randomly select umbrellas at a party, specifically focusing on the scenario where none of them picks their own umbrella. Participants explore methods to calculate the number of derangements for this situation.

Discussion Character

  • Exploratory, Technical explanation, Mathematical reasoning

Main Points Raised

  • One participant presents the problem of calculating the number of ways none of the friends take their own umbrella.
  • Another participant references a link to the "Hat-Check Problem," suggesting it may provide relevant insights or solutions.
  • A third participant identifies the problem as one of derangements and offers two methods for calculating the exact number: a recurrence relation and an alternating sum formula.

Areas of Agreement / Disagreement

Participants do not appear to reach a consensus on a specific solution, but they agree on the nature of the problem as one involving derangements.

Contextual Notes

The discussion does not clarify the assumptions behind the methods proposed or the applicability of the referenced link to the specific problem at hand.

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

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