Problem
COM-B1-M10-P010 Path Through a Point
#10
★★☆☆☆ Level 2 of 5
How many shortest paths from \((0,0)\) to \((5,4)\) pass through \((2,1)\)?
Multiply the number of paths before and after the point.
To \((2,1)\): \(\binom{3}{1}=3\). From \((2,1)\) to \((5,4)\): need \(3\) right and \(3\) up moves, so \(\binom{6}{3}=20\). Total: \(3\cdot20=60\).
A path through a fixed point splits independently.