The number of paths from (1, 1) to (8, 10) via (4, 6) = The number of paths from (1,1) to (4,6) * The number of paths from (4,6) to (8,10)
To calculate the number of paths from (1,1) to (4,6), 4-1 =3 steps in x-directions and 6-1=5 steps in y direction
Hence the number of paths from (1,1) to (4,6) = ${}^{(3+5)}C_3$ = 56
To calculate the number of paths from (4,6) to (8,10), 8-4 =4 steps in x-directions and 10-6=4 steps in y direction
Hence the number of paths from (4,6) to (8,10) = ${}^{(4+4)}C_4$ = 70
The number of paths from (1, 1) to (8, 10) via (4, 6) = 56*70=3920
