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Help on AIME Workshop Advanced Combo Q9

 
 
SunCaroline的头像
Help on AIME Workshop Advanced Combo Q9
SunCaroline - 2021年02月15日 Monday 17:09
 

I have tried a few methods for this problem, but each time my answer is different from the one in the answer key! May I get a hint or solution for how to solve this one?


(So far, the method I tried was to do case work based on how the ambassadors were arranged. For example, looking only at the even seats, there could be  1. 2 consecutive ambassadors, gap, two consecutive ambassadors, gap  OR   2. 3 consecutive ambassadors, gap, 1 ambassador, gap. Then for each case,  look at how many ways there are to place the advisors. I can't find anything wrong with this method but the answer's different!) 

 
LensmireJohn的头像
Re: Help on AIME Workshop Advanced Combo Q9
LensmireJohn - 2021年02月17日 Wednesday 13:30
 

These types of cases seem to be on the right track. However, you might be missing one case. Using $A$ for ambassador and $G$ for gap there should be three cases (up to rotation):

  1. $AAGAAG$
  2. $AAAGAG$
  3. $AAAAGG$

Then for each case you need to count the different rotations, arrange the ambassadors, and place the advisors. To help to make sure you're on the right track, case 1 should lead to $648$ different outcomes.

Hope this helps a bit!

 
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SunCaroline的头像
Re: Help on AIME Workshop Advanced Combo Q9
SunCaroline - 2021年02月19日 Friday 16:34
 

Hi! Thanks for the hint! I think I forgot case 3. Again, thank you so much!

 
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