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Math Challenge II-A Combinatorics Series

 
 
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Re: Math Challenge II-A Combinatorics Series
LensmireJohn - 2023年01月3日 Tuesday 11:02
 

I think there's one part of the wording of the original question that's a little confusing here. The phrase that is important is "exactly 9 students get A's on any two exams". That means if you pick any two exams, for example the 1st and 2nd exam, there are exactly 9 students who got an A on both. (Note further what the students get on the 3rd exam does not matter for this statement.)

Thus, try to fill in a Venn diagram with how many students got A's on each specific test:


For example, we know that in total "Test 1" should have 11 students somewhere in it's four sections, and the same for "Test 2" and "Test 3".

It looks like you were doing a slightly different question, where you were assuming:

  • exactly 11 students get A's on each exam
  • 9 students get A's on at least two exams (this is the one that isn't quite right)
  • 14 students get an A on at least one exam

Under these assumptions your work is good. HOWEVER, the question doesn't make sense in that scenario, because if 10 students get A's on at least 3 exams, that would mean that 10 (not 9) got A's on at least two exams, so things don't work.

Hope this helps!