Determine the number of paths that spell math

[F.B]

Imagine that the first traffic light you encounter on your way to school each morning has a 60 s cycle in which it is green for 20 s. What is the probability that you will get a green light on the next three morning trips to school?
I get the answer this way but i don't get why.
C(5,3) x (1/3)^3
I have two more questions they are similar but they are hard to post.

Since i can't really post this arrangement in a triangle i'll just describe it. There are 4 H's at the base of a triangle, then there are 3 T's above those and then 2 A's and then at the tip of the triangle there is an M.

Determine the number of paths that spell math, you have to start from the top and you can only move diagonally. I can figure out the answer my self which is 8 but i don't get why.
The reason why I am asking this is because there is a question similar to this where you have to spell mathematics, but its not in the shape of a triangle, its in the shape of a diamon sort of. So how would I work this out.

 

[Tide]

For your first problem, is the question asking for the probability of getting a green light on all three days or getting a green light exactly once over the three days? In any case, where does the 5 come from?

 

[F.B]

I thought it was in a week. But I'm not sure what the question is asking, i posted the exact question above.

 

[Tide]

The problem, as stated, asks for the probability over the course of 3 days. Your answer says the probability of getting a green light on all three days is 10/27. That is greater than 1/3 which is greater than the probability of getting a green light on any given day. Do you really believe that is possible?