  In a previous paper, we derived necessary and sufficient conditions for the
invertibility of square submatrices of the Pascal upper triangular matrix. To
do so, we established a connection with the two-point Birkhoff interpolation
problem. In this paper, we extend this result by deriving a formula for the
rank of submatrices of the Pascal matrix. Our formula works for both square and
non-square submatrices. We also provide bases for the row and column spaces of
these submatrices. Further, we apply our result to one-point lacunary
polynomial approximation.
