  This paper considers the $(n,k)$-Bernoulli--Laplace model in the case when
there are two urns, the total number of red and white balls is the same, and
the number of selections $k$ at each step is on the same asymptotic order as
the number of balls $n$ in each urn. Our main focus is on the large-time
behavior of the corresponding Markov chain tracking the number of red balls in
a given urn. Under reasonable assumptions on the asymptotic behavior of the
ratio $k/n$ as $n\rightarrow \infty$, cutoff in the total variation distance is
established. A cutoff window is also provided. These results, in particular,
partially resolve an open problem posed by Eskenazis and Nestoridi.

Probability;Combinatorics;Dynamical Systems