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January 8th, 2017, 06:01 PM  #1 
Senior Member Joined: Oct 2013 From: New York, USA Posts: 602 Thanks: 82  How Many Ways Can 9 People Be Made Into 3 Groups Of 3?
How many ways can 9 people be made into 3 groups of 3 without any two people being in the same group more than once?

January 8th, 2017, 07:22 PM  #2 
Senior Member Joined: Feb 2010 Posts: 679 Thanks: 127 
I'll take a stab and let others correct me if I'm wrong. I think it is $\displaystyle \dfrac{\binom{9}{3}\binom{6}{3}}{3!}=280$ 
January 9th, 2017, 04:19 AM  #3 
Senior Member Joined: Oct 2013 From: New York, USA Posts: 602 Thanks: 82  That's not what I'm looking for. Two people cannot be in the same group in multiple arrangements. Each person will be grouped with 2 of the other 8 people, so that limits it to 4 arrangements, but I don't know if the answer is 4 or less than 4.

January 9th, 2017, 02:46 PM  #4  
Senior Member Joined: Sep 2015 From: USA Posts: 1,976 Thanks: 1026  Quote:
$\{(1,2,3), (4,5,6), (7,8,9)\}$ $\{(1,4,7), (2,5,8 ), (3,6,9)\}$ $\{(1,5,9), (2,6,7), (3,4,8 )\}$ $\{(1,6,8 ), (2,4,9), (3,5,7)\}$ since each member shares exactly 1 group with every other member I don't think it's possible that there are any additional arrangements that can be added. it's possible that there is another set of 4 arrangements but 4 is going to be the maximum size of any set of valid arrangements.  
January 11th, 2017, 03:44 PM  #5 
Member Joined: Jan 2017 From: California Posts: 80 Thanks: 8 
Is there a formula for this?

January 11th, 2017, 04:36 PM  #6 
Senior Member Joined: Sep 2015 From: USA Posts: 1,976 Thanks: 1026  
January 11th, 2017, 05:56 PM  #7 
Senior Member Joined: Sep 2015 From: USA Posts: 1,976 Thanks: 1026  
January 11th, 2017, 05:59 PM  #8 
Member Joined: Jan 2017 From: California Posts: 80 Thanks: 8  

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