Online Course Discussion Forum
AMC 10 2013 A #17
Daphne is visited periodically by her three best friends: Alice, Beatrix, and Claire. Alice visits every third day, Beatrix visits every fourth day, and Claire visits every fifth day. All three friends visited Daphne yesterday. How many days of the next 
-day period will exactly two friends visit her?
I divided 365 by 12, 15, 20, 60 which is the lcm of 3and4, 3and5, and 4and5 and 3an4and5, and I got 30+24+18-6=66, but the correct answer is 54, I don't know why. I'm pretty sure it should A union B+B union C+C union A-A union B union C
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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