Pathways question?

Question

Ok this is an awful representation but Picture a square that is 5 x 5. The pont labelled S is a store, P is a pool, and T is a tennis Court. I have determined using Pascels Triangle that there are 192 paths from the Tennis Court to the Pool,

however part b) asks What percentage of the total paths can you get to P from T through S?

I believe it is 50% but I seem to remember in class there is actually some trick to this question.


_ . _ ._ ._ S_P
1_1_1_1_1_1
1_1_1_1_1_1
1_1_1_1_1_1
1_1_1_1_1_1
1_1_1_1_1_1
T

Answer 1

The path either has to go through S, or the point directly below P. You'll notice that they are precisely mirror images of each other - so they must have exactly the same number of paths through them.

(If you took one path through , and reflected it in the diagonal, you get a path that doesn't go through S, and vice versa).

So paths through S = paths not through S, so you get 50%.

Answer 2

i think it's 28