Ordering four relay legs among four runners

Question

A relay team has 4 runners, and the coach must assign them to the 4 legs of the race (leg 1, leg 2, leg 3, leg 4), one runner per leg. How many different lineups are possible?

Accepted Answer

Assigning distinct runners to ordered, distinct legs is the same as arranging 4 items in order. Leg 1 can be any of 4 runners, leg 2 any of the remaining 3, leg 3 any of 2, and leg 4 is then forced: 4 	imes 3 	imes 2 	imes 1 = 4! = 24. Because the legs are distinguishable (running first is different from running last), each way of matching runners to legs counts as its own lineup. Takeaway. Matching n distinct people to n distinct ordered slots is a permutation: n! ways. Four runners over four legs give 4! = 24.

Ordering four relay legs among four runners

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