7 is quickly solved going by the method given in the solution. I am going to try and make a more verbose essay out of it...
If you have got to stop at 3 stations, then 35 stations are left where you don't stop. Suppose you arrange these 35 stations in a row with a gap in between each: Visualised as _ X _ X _ .......X_ (each X denotes a station, there are 35 Xs with a gap before AND after each X)
There are a total of 36 gaps created. Observe that these gaps are such that if you choose any 2 gaps, there will be at least one station between them. Suppose you choose the first 3 gaps for your 3 mandatory stops, then your travel will look like:
Y X Y X Y XXX ... XX, where the Ys denote your stops and Xs denote the stations where you don't stop. The total no of ways of choosing 3 stops out of these 36 gaps is 36C3, as given in the solution. Don't know if this was any clearer than what was given there already!
In ques 20, there is a simple rule to be used. In a standard 12 hour clock, the hands make an angle of 180 (or 0) degrees 11 times in 12 hours. Any other angle is made 23 times in 12 hours. (Observation to be remembered)
Going by the same analogy, in a 9 hour clock, the hands will make an angle of 180 degrees 8 times in 9 hours. Between 9 pm on Monday and 9 am on Wednesday, the alien click covers 27 hours (9 hours till 9 am Tue, 18 hrs till 9 pm Tue and 27 hrs till 9 am Wed). 8 times in 9 hours means 24 times in 27 hours.