CAT 2019 Slot 1 QA Question & Solution
GeometryHard
Question
With rectangular axes of coordinates, the number of paths from (1, 1) to (8, 10) via (4, 6), where each step from any point (x, y) is either to (x, y+1) or to (x+1, y), is
Solution
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