CAT 2019 Slot 1 — QA Question 2
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
Answer & solution
Answer: 3920
We have to go from (1, 1) to (8, 10) via (4, 6)
Number of paths from (1,1) to (8,10) = [Number of paths from (1,1) to (4,6)] × [Number of paths from (4,6) to (8,10)]
Path from (1,1) to (4,6):
Number of horizontal displacements (âx) = 4 − 1 = 3 units and
Number of vertical displacements (ây) = 6 − 1 = 5 units.
Hence, a total of 8 units.
∴ Number of paths from (1,1) to (4,6) = 8C3 × 5C5 = 56.
Path from (4,6) to (8,10):
Number of horizontal displacements (âx) = 8 − 4 = 4 units and
Number of vertical displacements (ây) = 10 − 6 = 4 units.
Hence, a total of 8 units.
∴ Number of paths from (4,6) to (8,10) = 8C4 × 4C4 = 70.
Total required number of paths = 56 × 70 = 3920.
Hence, 3920.