10. On a particularly strange railway line, there is just one infinitely long track, so overtaking
is impossible. Any time a train catches up to the one in front of it, they link up to form a
single train moving at the speed of the slower train. At first, there are three equally spaced
trains, each moving at a different speed. After all the linking that will happen has happened,
how many trains are there? What would have happened if the three equally spaced trains
had started in a different order, but each train kept its same starting speed? On average
(where we are averaging over all possible orderings of the three trains), how many trains will
there be after a long time has elapsed? What if at the start there are 4 trains (all moving
at different speeds)? Or 5? Or n? (Assume the Earth is flat and extends infinitely far in all
directions.)
Answers
11/3 Trains
Step-by-step explanation:
Let say three trains
T1 , T2 & T3
if T1 , T2 & T3 is the order of train
and speed of T1 ≥ T2 ≥ T3
Then there will be 3 Trains as no linkage will be there
T1 , T3 & T2
speed of T2 ≥ T3
=> T2-T3 will link up but not with T1
so 2 Trains
T2 , T1 & T 3
as T1 speed ≥ T2
=> T2-T1 will link up but T3 Does not
so 2 Trains
T2 , T3 & T1
as T1 speed ≥ T3
=> T3-T1 will link up but T2 Does not
so 2 Trains
T3 , T1 , T2
T3-T1 will link up first then
speed of link will be speed of T3 hence T2 will also link up
so 1 Train
T3 T2 T1
All will link up
so 1 Train
1 Train in two case
2 Train in 3 cases
3 Train in 1 Case
Average = (1 * 2 + 2 * 3 + 3 * 1)/3 = (2 + 6 + 3)/3 = 11/3
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