We define a counting function that is related to the binomial coefficients. For this function, we derive an explicit expression. In some particular cases, we prove simpler explicit formulae. 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.

We then examine the relationship of our counting function with different classes of integers. These classes include: some 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 Delannoy 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 count several configurations by our function. Some of these are: the number of spanning subgraphs of the complete bipartite graph, the number of squares contained in a square, the number of colorings 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.

For the most statements in the paper, we provide bijective proofs in terms of insets, which we define in the paper. Hence, using the same method, we count different configurations.