Lattice paths to (5,3) staying below the diagonal

A path on the integer lattice begins at $(0,0)$ and ends at $(5,3)$, and each step moves one unit either rightward or upward. The path may touch the line $y=x$ but must never go above it. How many such paths exist?

$28$. Without restriction there are $\binom83=56$ paths. A path that goes above $y=x$ touches $y=x+1$; reflecting its start across that line maps it bijectively to a path from $(-1,1)$ to $(5,3)$, of which there are $\binom82=28$. So $56-28=28$ paths are valid.