We know Adam must move North 6 blocks and West 4 blocks. Consider his moves. In all he must make 10 moves represented by NNNNNNWWWW. But we can have any permutation of these.
The number of such permutations is:
10!/(6!*4!)=10*9*8*7*6*5*4*3*2/((6*5*4*3*2)*(4*3*2))=10*9*8*7/(4*3*2)=210.
Since there are only 195 school days in the year he won't be able to cover all paths.