wu :: forums - Print Page


    
      
        wu :: forums
        (http://www.ocf.berkeley.edu/~wwu/cgi-bin/yabb/YaBB.cgi)
      

        riddles >> medium >> Foot Buddies
        
(Message started by: ThudanBlunder on Jan 20^th, 2009, 6:32pm)

Title: Foot Buddies
Post by ThudanBlunder on Jan 20^th, 2009, 6:32pm

Six colleges each send their two best runners to an athletics meeting. The 12 runners are randomly assigned to the six available lanes, with two runners in each lane. What is the probability that at least one college will have both of its runners assigned to the same lane?

Title: Re: Foot Buddies
Post by balakrishnan on Jan 21^st, 2009, 2:14am

The number of ways in which 2N runners(2 from N teams each) shall be distributed in the N lanes such that atleast one college has both its runners assigned to the same lane is given by(using simple [hide]inclusion-exclusion[/hide])
[hide]sum(i=1,N,(-1)^(i-1)*(binomial(N,i))^2*i!*2^i*(2*N-2*i)!)[/hide]
and hence the probability is
[hide]sum(i=1,N,(-1)^(i-1)*(binomial(N,i))^2*i!*2^i*(2*N-2*i)!)/(2*N)![/hide]

For N=6, the probability happens to be [hide]871/2079[/hide]