Problem
COM-B2-M03-P008 Path Through a Point
#8
★★★☆☆ Level 3 of 5
How many monotone paths from \((0,0)\) to \((7,5)\) pass through \((3,2)\)?
Split the path into two independent parts.
The path to \((3,2)\) contains \(3\) steps \(R\) and \(2\) steps \(U\), so there are \(\binom{5}{2}=10\) choices. From \((3,2)\) to \((7,5)\), we need \(4\) steps \(R\) and \(3\) steps \(U\), so \(\binom{7}{3}=35\) choices. The parts are independent, so the answer is \(10\cdot35=350\).
Useful for later path-decomposition problems.