Grid Paths via an Intermediate Point

CAT 2019 Slot 1 · QA · Hard · Co-geometry

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With rectangular axes of coordinates, the number of paths from (1,1)(1,1) to (8,10)(8,10) via (4,6)(4,6), where each step from any point (x,y)(x, y) is either to (x,y+1)(x, y+1) or to (x+1,y)(x+1, y), is

Answer

3920

Explanation

Path from (1,1)(1,1) to (4,6)(4,6): right steps =3= 3, up steps =5= 5. Number of paths =(3+53)=(83)=56= \binom{3+5}{3} = \binom{8}{3} = 56. Path from (4,6)(4,6) to (8,10)(8,10): right steps =4= 4, up steps =4= 4. Number of paths =(4+44)=(84)=70= \binom{4+4}{4} = \binom{8}{4} = 70. Total paths =56×70=3920= 56 \times 70 = 3920.

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