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How many ways are there to divide 12 people into a group of 3, a group of 4, and a group of 5, if Henry has to be in the group of 4?

 Sep 6, 2020
 #1
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There are 84,320 ways to divide the 12 people.

 Sep 6, 2020
 #2
avatar+110689 
+2

I am not sure but this is my best guess.

 

Forget about Henry, there are 11 people which are to be divided into groups of 3,3, and 5

 

There are 11C5 ways to choose the group of 5

Now there are 6C3 ways to choose a group of 3 BUT this is double counting, I need to divide by 2

 

so I have    11C5 * (6C3 /2)

 

Now there are 2 options for where to put Henry.

 

so I now have    11C5 * (6C3 /2) * 2 = 462*20/2*2 = 9240 option.

 Sep 7, 2020
 #3
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+1

Melody's solution is accurate. It is the same as: 11! /(5! * 3! * 3!) =9,240 ways. Henry can sit with any group of 3.

 Sep 8, 2020
 #4
avatar+110689 
+2

ok, thanks   But I would prefer to describe it as

 

11! /(5! * 3! * 3!)   divide by 2 because the 3! and 3! are interchangeable. multiply by 2 because there are 2 ways to add in Henry.

 

=  9,240 ways

 

 

I am still having some trouble thinking about it this way though. 

This would normally be the answer to a question like this.

How many ways can the 11 letters  aaabbbcccc be arranged in a line.   11!/(3!3!4!)  is the answer.    [I am not going to concern myself with Henry]

 

For our question it would be how many ways can 11 objects be put in a line if the order of the first 5 does not matte, the order of the next 3 does not matter and the order ot the last 3 does not matter AND the second and third groups of 3 can be swapped around. 

NOW it does make sense to me.    11!/(3!3!4!) divided by 2

 

Thanks guest for making me think about that.

Melody  Sep 9, 2020
edited by Melody  Sep 9, 2020
edited by Melody  Sep 9, 2020
 #5
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Thank you very much! But 9420 is incorrect

 Sep 10, 2020
edited by HelpBot  Sep 10, 2020
 #6
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You say " Thank you very much! But 9420 is incorrect "!, But the answer given to you is 9240 and NOT 9420 ?!.

 Sep 11, 2020
 #7
avatar+239 
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i'm sorry about that, that was a typo I had

 Sep 13, 2020
 #8
avatar+110689 
0

You have not made your meaning clear at all. 

Your last two comments have both been totally without meaning.

 

Do you think the answer given is incorrect?

Why do you think this?  Give your reason. 

Do you know what the correct answer is said to be? Do you know why?

Melody  Sep 15, 2020

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