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?
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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 ! combinations!) ( 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 ! for these 5 sprinter.
So the possible assignments in this case is 3 × 2 × 6 × 2 ! × 5 ! = 8 6 4 0
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 ! combinations!) ( 2 , 4 , 6 ) , ( 2 , 4 , 7 ) , ( 2 , 5 , 7 ) , ( 3 , 5 , 7 )
Again, the remaining 5 sprinters yield 5 ! combinations on the 5 remaining lanes.
So the possible assignments in this case is 4 × 3 ! × 5 ! = 2 8 8 0
At last, add up all possible assignments, we have 8 6 4 0 + 2 8 8 0 = 1 1 5 2 0