Question

Problems 6a and 6b. Please explain how you got your answer.

6. T he 12 members of a cross country team show up for preseason before the beginning of classes. They are rooming together in pairs. (a) In how many ways can 6 pairs be formed from 12? (For simplicity, let's assume the pairs are numbered 1 to 6. Ask yourself: in how many ways can the first pair be assigned? The second? Etc.)

Answer 1

a)(number of ways =N(select 2 from 12 for first pair then select 2 from 10 for 2nd pair and so on)

=¹²C₂*¹⁰C₂*⁸C₂*⁶C₂*⁴C₂*²C₂ /6! =12!/(2!*2!*2!*2!*2!*2!*6!)=10395 ( here we have divided by 6! as order of pairs does not matter)

b)

number of ways 12 runners can be arranged =10395

number of ways that none of white and black pair exist)=N(ways 6 black player are arranged with in them)*N(6 white player are arranged with in them)

=(6!/(2!*2!*2!*3!))*(6!/(2!*2!*2!*3!)) =15*15=225

hence probability =225/10395=5/231