Derangements with probability
Source: Canadian Repêchage 2013: Problem 4
May 20, 2014
countingderangementprobabilitycombinatorics proposedcombinatorics
Problem Statement
Four boys and four girls each bring one gift to a Christmas gift exchange. On a sheet of paper, each boy randomly writes down the name of one girl, and each girl randomly writes down the name of one boy. At the same time, each person passes their gift to the person whose name is written on their sheet. Determine the probability that both of these events occur:[*] (i) Each person receives exactly one gift;[*] (ii) No two people exchanged presents with each other (i.e., if gave his gift to , then did not give her gift to ).