Counting Problem Homework: 100 People into 10 Grps of 10

In summary, there are (10!)^11 ways to divide 100 people into 10 discussion groups with 10 people in each group, taking into account the different ways to rearrange within columns and the columns themselves. This accounts for the repeated counts and ensures that the groups are distinct.
  • #1
cragar
2,552
3

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.
 
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  • #2
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...
 
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  • #3
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
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
thanks for your responses, you i think my answer makes sense.
 

Related to Counting Problem Homework: 100 People into 10 Grps of 10

1. How many different ways can 100 people be divided into 10 groups of 10?

The number of different ways to divide 100 people into 10 groups of 10 is equal to the number of different combinations, which can be calculated using the formula nCr = n! / r!(n-r)!, where n is the total number of people (100) and r is the number of groups (10). This results in 99,884,400 different ways to divide the 100 people into 10 groups of 10.

2. Is there a specific method or strategy to use when dividing 100 people into 10 groups of 10?

Yes, there are several methods and strategies that can be used to divide 100 people into 10 groups of 10. One method is to randomly assign each person to a group, making sure that each group has an equal number of people. Another method is to divide the people based on certain characteristics, such as age, gender, or skill level. Ultimately, the method used will depend on the purpose of the groups and the preferences of the person doing the dividing.

3. Can 100 people be divided into 10 groups of 10 if some people have to be in specific groups?

Yes, it is possible to divide 100 people into 10 groups of 10 if some people have to be in specific groups. This can be achieved by first placing the specific people in their designated groups, and then dividing the remaining people into the remaining groups. Alternatively, the specific people can be distributed evenly among the remaining groups, as long as there are enough people to fill each group.

4. What is the purpose of dividing 100 people into 10 groups of 10?

The purpose of dividing 100 people into 10 groups of 10 can vary depending on the context. For example, it could be for a team-building exercise, a group project, or for seating arrangements at an event. Dividing into smaller groups can also make it easier for people to interact and work together, as well as help with organization and management.

5. Is there a limit to the number of people that can be divided into 10 groups of 10?

No, there is no limit to the number of people that can be divided into 10 groups of 10. As long as there are at least 10 people, they can be divided into 10 groups of 10 or more. However, as the number of people increases, the number of possible combinations also increases, making it more challenging to divide them evenly into 10 groups of 10.

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