On a 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. You watch, and eventually (after all
the linking that will happen has happened), you count the trains. You wonder what
would have happened if the trains had started in a different order (but each of the
original three trains had kept its same starting speed). On average (averaging over all
possible orderings), 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?

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

Learn more:

a car is moving at an average speed of 161/3 km/hr for 1 and 1/2 hrs ...

The ratio Between the speed of a truck car and train in 3:8:9.The car ...

The total amount of trains remaining is equal to the total amount of starting trains that are slower than every train before them (including the first train)

Step-by-step explanation:

Lets do this exercise directly with n trains.

The first train will 'fuse' with the train immediately behind him only if that train is faster. If it is slower, it wont never catch up because the first train will mantain its speed and the second train can only mantain it by fusing itself with a faster train behind it (the speed of the fused train will be the speed of the slower of the 2 trains fused), thus it wont ever adquire the needed speed to catch the first train.

Now, if the first train is slower than the second train, the second train cant lose speed by fusing with trains below him (same argument as above), but since it is faster than the first train, it will eventually fuse with it and the speed will be reduced to the speed of the first train, giving the possibility for other trains to fuse with them too. In short, all consecutive trains starting from the second one and faster than the first train will fuse with the first train.

 The first train slower than the first train wont fuse with this 'chain' of trains, and it will start its own 'chain' of trains, the same way the first train did (it will fuse with any consecutive train faster that it starting from its following train).

This shows that each train slower than the previous trains will form its own chain of trains and it will stay forever. Thus, in the end, the total amount of trains remaining is equal to the total amount of starting trains that are slower than every train before them (including the first train).