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Probability/Combinatorics Question

  1. Sep 17, 2014 #1
    1. The problem statement, all variables and given/known data

    The NHL currently has a total of 30 teams in 4 divisions (7 teams in the Pacific
    Division, 7 in the Central Division, 8 in the Metropolitan Division, and 8 in the
    Atlantic Division). Suppose the NHL gets a new commissioner, and they have
    the curious notion of reshuffling teams by randomly assigning the 30 teams to the
    divisions (leaving the number of teams in each division the same as above). How
    many different ways can this be done?

    2. Relevant equations

    None.

    3. The attempt at a solution

    There are 30 teams and they must be partitioned into teams of 7, 7, 8, 8, which represent the four divisions..

    [tex] (30!)/(7!7!8!8!) [/tex] using a partitioning rule..

    I'm not sure if this is the right way to go about it, any hints would be appreciated.
     
  2. jcsd
  3. Sep 17, 2014 #2

    Ray Vickson

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    Science Advisor
    Homework Helper

    Sure, and to convince yourself you can do it sequentially. Call the divisions 1--4. In how many distinct ways can you assign teams to division 1? (That is, we pick the 7 to go into division 1 and the remaining 23 go into non-1.) For each distinct division-1 assignment, in how many different ways can we assign 7 to division 2? (That is, of the 23 still left, we assign 7 to division 2 and the other 16 to not [1 or 2].) The first division can be picked in
    [tex] N_1 = \binom{30}{7} = \frac{30!}{7! \; 23!} [/tex]
    different ways. For each such assignment the second division can be picked in
    [tex] N_2 = \binom{23}{7} = \frac{23!}{7! \; 16!} [/tex]
    different ways. Together, divisions 1 and 2 can be assigned in
    [tex] N_{12} = N_1 \: N_2 = \frac{30!}{7! \; 23!} \cdot \frac{23!}{7! \; 16!}
    = \frac{30!}{ 7! \; 7! \; 16!} [/tex]
    different ways. Keep going like that until all divisions have been assigned, and you will get your suggested solution.
     
  4. Sep 17, 2014 #3
    Thanks for the reassurance, I appreciate it. I don't have much experience in the math stats setting as I recently switched into the Statistics major, but I think I'm getting the hang of it.
     
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