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# Advanced Math/probability puzzle

Question
There was a race going on between 8 persons. In
how many ways, counting the ties, they can
reach
the destination i.e. finishing line...

This is either a very easy problem or a very hard problem depending how you account for ties.

For a high enough resolution timing system, there won't be any ties since no 2 racers can arrive at exactly the same time (probability of set of has measure zero). In this case, the number of ways they can reach the finish line = 8! (note the factorial, !, notation). FYI, this is obtained by considering the following:

8 ways to pick the 1st finisher from the group of 8,
7 ways to pick the 2nd, so for the 1st 2 choices, there are 8･7 = 56 ways to choose the finishers,
6 ways to pick the 3rd,
etc. until the last, finisher, so that number of ways = 8! = 8･7･6･5･4･3･2･1 = 40,320.

Now, the only way to include ties in the calculation is to assume a certain small time interval, ∆t, that represents the resolution of the timing system. If 2 (or more) finishers cross the finish line within this interval, they are considered tied. Note that if this happens, then the order of their finish is ambiguous and this must be taken into account in counting the ways they can finish. Complicated but not impossible.

Let me know if you are interested in this case.

Randy

Volunteer

#### randy patton

##### Expertise

college mathematics, applied math, advanced calculus, complex analysis, linear and abstract algebra, probability theory, signal processing, undergraduate physics, physical oceanography

##### Experience

26 years as a professional scientist conducting academic quality research on mostly classified projects involving math/physics modeling and simulation, data analysis and signal processing, instrument development; often ocean related

Publications
J. Physical Oceanography, 1984 "A Numerical Model for Low-Frequency Equatorial Dynamics", with M. Cane

Education/Credentials
M.S. MIT Physical Oceanography, B.S. UC Berkeley Applied Math

Past/Present Clients
Also an Expert in Oceanography