  In this paper, we study $F_{n}(x,k)$, the number of binary strings of length
$n$ containing $x$ zeros and a longest subword of $k$ zeros. A recurrence
relation for $F_{n}(x,k)$ is derived. We expressed few known numbers like
Fibonacci, triangular, number of binary strings of length $n$ without $r$-runs
of ones and number of compositions of $n+1$ with largest summand $k+1$ in terms
of $F_{n}(x,k).$ Similar results and applications were obtained for
\^F$_{n}(x,k),$ the number of all palindromic binary strings of length $n$
containing $x$ zeros and longest $k$-runs of zeros.
