Finishing orders in a seven-runner race

Question

Seven sprinters run a race with no ties possible. How many different finishing orders (from first to last) can there be?

Accepted Answer

A finishing order is just an arrangement of the 7 runners. Any of the 7 could finish first, then 6 remain for second place, 5 for third, and so on: 7 	imes 6 	imes 5 	imes 4 	imes 3 	imes 2 	imes 1 = 7! = 5040. Since no two runners tie, each ranking is a clean ordering of all seven, so the count is the factorial of the number of runners. Takeaway. With no ties, ranking n competitors is the same as ordering n distinct items: n!. Seven runners give 7! = 5040.

Finishing orders in a seven-runner race

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