Online Course Discussion Forum
AMC 10 2013 A #17
The overlaps are a little complicated here and we have to be careful with our overcounting in this case. Your numbers are all correct, but let's look at them a little more carefully:
- 30 days where A and B visit
- 24 days where A and C visit
- 18 days where B and C visit
- 6 days where A, B, and C visit
We want to count the number of days where exactly 2 of the 3 visit, so we start with 30+24+18, but this is overcounting.
For the overcounting, note that if A, B, and C visit all visit (situation #4 above), that is also counted in situations #1, #2, and #3. Thus, situation #4 is overcounting 6 times when we're counting situation #1, but also 6 times in situation #2 and #3.
Hence, we actually need to subtract 3*6 = 18 from 30+24+18 to get a final answer of 54. Hope this helps!
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