  The lattice of monotone triangles $(\mathfrak{M}_n,\le)$ ordered by
entry-wise comparisons is studied. Let $\tau_{\min}$ denote the unique minimal
element in this lattice, and $\tau_{\max}$ the unique maximum. The number of
$r$-tuples of monotone triangles $(\tau_1,\ldots,\tau_r)$ with minimal infimum
$\tau_{\min}$ (maximal supremum $\tau_{\max}$, resp.) is shown to
asymptotically approach $r|\mathfrak{M}_n|^{r-1}$ as $n \to \infty$. Thus, with
high probability this event implies that one of the $\tau_i$ is $\tau_{\min}$
($\tau_{\max}$, resp.). Higher-order error terms are also discussed.
