A group of 60 students is to be randomly assigned to two classes of 30 each. (the random assignment is ordered by the school district to ensure against any bias). five of the second graders M, S, K, R, C are closed friends. what is the probability that they will all be in the same class? what is the probability that exactly four of them will be? what is the probability that M will be in one class and her friends in other class?

• we have two groups: 5 close friends and 55 others
  1)
  prob of of ways to pick a class: 2 ways
  prob of ways to pick the five friends: 1
  prob of ways to pick 5 from 60: 60C5
  P(all 5 are in the same class): 2/60C5 = 0.00000036
  
  2)probability that exactly four of them will be
  of ways to pick a class: 2
  prob of ways to pick 4 from the five friends: 5C4 = 5
  # of ways to pick 1 from 55: 55
  # of ways to pick 5 from 60: 60c5
  P(exactly 4 of the 5 friends are in one of the classes)= [2*5*55]/60c5
  = 550/5461512 = 0.0001007047...
  
  3)probability that M will be in one class and her friends in other class
  prob of ways to pick a class for M : 2
  prob of ways to pick the remaining 29 in her class: 55C29 = 3.560597x10^15
  prob of ways to pick 30 from 60: 1.182646x10^17
  P(M in one class and friends in the other) [2*3.560597x10^15]/1.182646x10^17
  = 0.060214
  
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