Online Course Discussion Forum
Help on AIME Workshop Advanced Combo Q9
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!)
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):
- $AAGAAG$
- $AAAGAG$
- $AAAAGG$
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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