  We define a counting function that is related to the binomial coefficients.
  An explicit formula for this function is proved.
  In some particular cases, simpler explicit formuls are derived. We also
derive a formula for the number of (0,1)-matrices, having a fixed number of
1's, and having no zero rows and zero columns.
  Further, we show that our function satisfies several recurrence relations.
  The relationship of our counting function with different classes of integers
is then examined. These classes include: different kind of figurate numbers,
the number of points on the surface of a square pyramid, the magic constants,
the truncated square numbers, the coefficients of the Chebyshev polynomials,
the Catalan numbers, the Dellanoy numbers, the Sulanke numbers, the numbers of
the coordination sequences, and the number of the crystal ball sequences of a
cubic lattice.
  In the last part of the paper, we prove that several configurations are
counted by our function. Some of these are: the number of spanning subgraphs of
the complete bipartite graph, the number of square containing in a square, the
number of coloring's of points on a line, the number of divisors of some
particular numbers, the number of all parts in the compositions of an integer,
the numbers of the weak compositions of integers, and the number of particular
lattice paths.
  We conclude by counting the number of possible moves of the rook, bishop, and
queen on a chessboard.
  The most statements in the paper are provided by bijective proofs in terms of
insets, which are defined in the paper. With this we want to show that
different configurations may be counted by the same method.

Combinatorics