100 m Sprint

The following 8 sprinters will participate in a fictitious 100 m sprint:
Usain Bolt: Jamaica,
Yohan Blake:Jamaica,
Asafa Powell :Jamaica,
Tyson Gray :USA,
Richard Thompson :Trinidad and Tobago,
Donovan Bailey: Canada,
Olusoji Fasuba: Nigeria, and
Frankie Fredericks: Namibia

If the TWO rules below need to be followed, how many different lane assignments are possible?

  1. NO two sprinters from the same country can be in adjacent lanes.
  2. The outside lanes ( 1 and 8) can't be occupied by sprinters from the same country.


The answer is 11520.

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1 solution

Raymond Chan
Jun 6, 2018

There are 2 cases we have to consider: one of the outside lane is occupied by a Jamaica sprinter and all Jamaica sprinters are on the inside lanes.

Case 1: one of the outside lane is occupied by a Jamaica sprinter

From the 3 Jamaica sprinters, we choose 1 to stand on a outside lane (Let's say, lane 1) (the number of cases for lane 8 is the same, so just times 2 finally).

Then the other 2 Jamaica sprinters can only stand on lane 3-7 due to the rules, and they cannot stand adjacent to each others, which yield 6 possible arrangements for them (remember, each arrangement yields 2 ! 2! combinations!) ( 3 , 5 ) , ( 3 , 6 ) , ( 3 , 7 ) , ( 4 , 6 ) , ( 4 , 7 ) , ( 5 , 7 ) (3, 5), (3, 6), (3, 7), (4, 6), (4, 7), (5, 7)

For the rest of 5 sprinter, just arrange them randomly on the 5 remaining lanes. Therefore there are 5 ! 5! for these 5 sprinter.

So the possible assignments in this case is 3 × 2 × 6 × 2 ! × 5 ! = 8640 3 \times 2 \times 6 \times 2! \times 5! = 8640

Case 2: all Jamaica sprinters are on the inside lanes

All Jamaica sprinters can only stand on lane 2-7 without "adjacenting" to each other, which only yields 4 possible arrangements for them (remember, each arrangement yields 3 ! 3! combinations!) ( 2 , 4 , 6 ) , ( 2 , 4 , 7 ) , ( 2 , 5 , 7 ) , ( 3 , 5 , 7 ) (2, 4, 6), (2, 4, 7), (2, 5, 7), (3, 5, 7)

Again, the remaining 5 sprinters yield 5 ! 5! combinations on the 5 remaining lanes.

So the possible assignments in this case is 4 × 3 ! × 5 ! = 2880 4 \times 3! \times 5! = 2880

At last, add up all possible assignments, we have 8640 + 2880 = 11520 8640+2880=\boxed{11520}

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