You are to go from a to b, always moving right or down along the lines. on how many different a paths can you go?

We could fill those paths like this:
        1        1       1       1
1      2        3       4       5

1      3        6       10     15
1      4       10      20     35
         Or we can calculate:  3C7  = ( 7 * 6 * 5 ) / ( 3 * 2 * 1 ) = 35    You can go on 35 different paths.