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March 22nd, 2010, 07:02 PM  #1 
Senior Member Joined: Sep 2009 Posts: 115 Thanks: 0  choosing question
A student council consists of three freshmen, four sophomores, three juniors, and five seniors. How many committees of eight members can be formed containing at least one member from each class? so we know the total number of ways is : 16 choose 8. Then I am not sure if this is right but do we find the number that contain none from each class such as: none from freshmen would be 13 choose 8 none from sophomores would be 12 choose 8 none from juniors would be 13 choose 8 none from seniors would be 11 choose 8 Hence, we subtract now: [16 choose 8]  ([13 choose 8] + [12 choose 8] + [13 choose 8] + [11 choose 8]) I am not sure if this is right. Thank you in advance. 
March 22nd, 2010, 07:45 PM  #2 
Global Moderator Joined: Nov 2006 From: UTC 5 Posts: 16,046 Thanks: 938 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms  Re: choosing question
It's not quite right, because you've subtracted off (for example) councils with only freshmen and sophomores twice.

March 23rd, 2010, 09:09 AM  #3  
Math Team Joined: Dec 2006 From: Lexington, MA Posts: 3,267 Thanks: 408  Re: choosing question Hello, wannabe1! This is more complicated than you or I thought. Quote:
[color=beige]. . [/color] [color=beige]. . [/color] [color=beige]. . [/color] [color=beige]. . [/color] [color=beige]. . [/color] [color=beige]. . [/color] [color=beige]. . [/color] Well, I hope my reasoning is correct! [color=beige] .[/color]  
March 23rd, 2010, 09:55 AM  #4 
Global Moderator Joined: Nov 2006 From: UTC 5 Posts: 16,046 Thanks: 938 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms  Re: choosing question
Generally, you;d have to continue with the number of councils with no Fs, Ss, or Js; the councils with no Fs, Ss, or Srs; the councils with no Fs, Js, or Srs; the councils with no Ss, Js, or Srs. But in this case there are none, so the answer looks good to me. Lookup "inclusionexclusion" for more. 
March 23rd, 2010, 04:24 PM  #5 
Senior Member Joined: Sep 2009 Posts: 115 Thanks: 0  Re: choosing question
o ya your right, thanks that is really helpful (ps. I was doing it with 4 juniors that is why i chose from 16 members) and you are correct about the overlap I totally forgot. Thank you again. 

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