Certain number of people arranged in several groups

[desmond iking]

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

 

[Fredrik]

This is only a matter of choosing three people that you tell to step away from the others.

 

[desmond iking]

This is only a matter of choosing three people that you tell to step away from the others.

can you explain further?

 

[Fredrik]

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).

 

[desmond iking]

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.

 

[Fredrik]

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.