Counting problem

  • Thread starter cragar
  • Start date
  • #1
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Homework Statement



100 hundred people are to be divided into 10 discussion groups with 10 people in each group
how many ways can this be done.

The Attempt at a Solution


So if we think of it as people on a 10 by 10 grid their are 100! ways of populating the grid and then 10! ways or rearranging the columns and 10! ways of rearranging the rows.
so would the answer be
[itex] \frac{100!}{10!^{10}10!} [/itex]
I have 10!^10 because there are ten columns and each of those 10 columns can be arranged 10! ways. And the other 10! on the bottom because I could rearrange those rows 10! ways. And I divide 100! by them because we are over counting. If we switch the people around in the group it is still a distinct group.
 

Answers and Replies

  • #2
lanedance
Homework Helper
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didn't totally follow the re-arranging of the cols, so as another way for the first group we choose 10 from 100 without order
[tex] \frac{100!}{10!90!} [/tex]

for the 2nd group we choose 10 from 90 without order
[tex] \frac{90!}{10!80!} [/tex]

so putting these together for both groups we get
[tex] \frac{100!}{10!90!} \frac{90!}{10!80!} = \frac{100!}{10!^280!} [/tex]

and so on, once finished counting the groups we need to account for the different ways to arrange ten groups which represents the repeated counts -

but its starting to look pretty similar to your answer...
 
Last edited:
  • #3
lanedance
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also another check would be to see if you formula generalise to n^2 people in n groups and check for n = 2 and n= 3
 
  • #4
lanedance
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ok - so i get your idea now, but think it needs a little tweek.. the rows got me

100! ways to populate a grid, where the columns represent a group
- 10! ways to rearrange within a single column and 10 columns gives a total (10!)^10 ways to rearrange within columns
- there is also 10! different ways to arrange the columns themsleves
So in all there is (10!)^11 ways to rearrange the grid for a given combination that leads to the same group structure
 
Last edited:
  • #5
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thanks for your responses, ya i think my answer makes sense.
 

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