Certain number of people arranged in several groups

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In summary: So, can you figure out how to use that idea to correctly solve the problem?In summary, the question is asking for the number of ways 9 people can be divided into two groups of 6 and 3 people. One approach to solving this is by considering the number of ways to divide the 9 people into two groups of 3 and 6, and then accounting for the fact that the smaller group of 3 can be placed on either the left or right side of the larger group without changing the overall grouping. This results in a total of 8640 possible arrangements. However, the given answer is 84, which can be explained by considering that it is simply a matter of choosing 3 people to step away from
  • #1
desmond iking
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Homework Statement


Find the number of ways of 9 people can be divided into two groups of 6 people and 3 people.


Homework Equations





The Attempt at a Solution



my working is there are 6! arrangement for 6 people and 3! arrangment for 3 people. then the group of 3 people can be placed at right or left only . so that the 6 people form another group will not be seperated. so my working = 3! x 6! x2 = 8640..

the ans given is only 84
 
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  • #2
This is only a matter of choosing three people that you tell to step away from the others.
 
  • #3
Fredrik said:
This is only a matter of choosing three people that you tell to step away from the others.

can you explain further?
 
  • #4
desmond iking said:
can you explain further?
I can't say much more without completely solving the problem for you. Would you agree that if you move 3 people away from the others, you have divided the original group into two, one with 3 people and one with 6? This is of course not the only way to do it. You don't have to separate them physically. You can e.g. hand out funny hats to 3 of them. But no matter what you do, it's a matter of choosing 3 people (or 6 people).
 
  • #5
No i mean I move the 3 people in a group as 1 item in the arrangement. And the same thing for the 6 people.
 
  • #6
I don't quite understand what you mean by that. The fact that your calculation includes 6! and 3! suggests that you're taking into account the number of ways that each group can be rearranged, but you're not doing it correctly. How many ways are there to divide 3 people into two groups of 1 and 2? Your method yields 2!·1!·2=4, but the right answer is clearly 3. How many ways are there to divide 4 people into two groups of 2 each? Your method yields 2!·2!·2=8, but the correct answer is 6. That's the number of ways you can distribute 2 identical funny hats among 4 people.
 

Related to Certain number of people arranged in several groups

1. How do you determine the number of possible arrangements of a certain number of people into several groups?

The number of possible arrangements can be calculated by using the formula nCr = n!/(r!(n-r)!), where n is the total number of people and r is the number of groups.

2. Are there any restrictions or rules for arranging people into groups?

Yes, there are often restrictions or rules that must be followed, such as a minimum or maximum number of people in each group, or certain people who must be in the same or different groups.

3. How do you ensure fairness and equality in the arrangement of people into groups?

To ensure fairness and equality, randomization can be used to assign people to different groups. This helps to avoid any biases or preferences in the arrangement.

4. Can the arrangement of people into groups affect the outcome of a study or experiment?

Yes, the arrangement of people into groups can significantly impact the outcome of a study or experiment. Different groupings can lead to different results and conclusions.

5. Are there any practical applications for arranging people into groups?

Yes, arranging people into groups is commonly used in various fields such as market research, clinical trials, and social studies to analyze and understand human behavior and patterns.

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