A combinatorial proof of a formula for Betti numbers of a stacked polytope

For a simplicial complex $\mathrm{\Delta }$, the graded Betti number ${\beta }_{i,j}(k[\mathrm{\Delta }])$ of the Stanley-Reisner ring $k[\mathrm{\Delta }]$ over a field $k$ has a combinatorial interpretation due to Hochster. Terai and Hibi showed that if $\mathrm{\Delta }$ is the boundary complex of a $d$-dimensional stacked polytope with $n$ vertices for $d\ge 3$, then ${\beta }_{k-1,k}(k[\mathrm{\Delta }])=(k-1)\left(\genfrac{}{}{0px}{}{n-d}{k}\right)$. We prove this combinatorially.