  The numbers we study in this paper are of the form $B_{n, p}(k)$, which is
the number of binary words of length $n$ that contain the word $p$ (as a
subsequence) exactly $k$ times. Our motivation comes from the analogous study
of pattern containment in permutations. In our first set of results, we obtain
explicit expressions for $B_{n, p}(k)$ for small values of $k$. We then focus
on words $p$ with at most $3$ runs and study the maximum number of occurrences
of $p$ a word of length $n$ can have. We also study the internal zeros in the
sequence $(B_{n, p}(k))_{k \geq 0}$ for fixed $n$ and discuss the unimodality
and log-concavity of such sequences.
