Re: Difference between two symmetric combinatoric questions?
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07 May 2019, 07:47
Everything you're saying is true (so you might not be as confused as you think!), but the questions are different, though in a way that might be easy to miss. In the second question, we have 5 different people running the race. So the order of the three other people (not John and Mike) will matter - we'd consider John, Mike, Amir, Belinda, Carlos to be different from John, Mike, Carlos, Amir, Belinda, for example. But in the first question, we don't have 5 people -- we only have the two, Bob and Rachel. And in that case, Bob, Rachel, Empty1, Empty2, Empty3 is the same seating arrangement as Bob, Rachel, Empty3, Empty1, Empty2, because in each case, the last three chairs are empty.
There are some cumbersome solutions to these problems in the links you mentioned. For the footrace question, 5 people can finish in 5*4*3*2*1 = 120 orders. If the finish order is random, half the time John beats Mike, and half the time Mike beats John, so in half of these arrangements, John is ahead of Mike, and the answer is 60.
For the seating question, if we had no restrictions, we'd have 5 choices for where to seat Bob, then 4 choices for where to seat Rachel, for 5*4 = 20 possible seating arrangements. In half of these arrangements, Bob is left of Rachel, and in the other half, Rachel is left of Bob, so in 10 of them Bob is on the left.
Notice it's the additional "3*2*1" in the first solution, the arrangements of the remaining 3 people, that lead to the difference in the answers - that's the reason one answer is 6 times the other.