Problem
COM-B1-M10-P016 Path Avoiding a Point
#16
★★★☆☆ Level 3 of 5
How many shortest paths from \((0,0)\) to \((5,4)\) do not pass through \((2,1)\)?
Subtract the paths passing through that point.
There are \(\binom{9}{5}=126\) paths in total. Paths through \((2,1)\) number \(\binom{3}{1}\cdot\binom{6}{3}=3\cdot20=60\). Hence \(126-60=66\) paths avoid the point.
Combines paths and complement counting.