We consider $m\times n$, $m\le n$, matrices with entries from an arbitrary given finite set of nonnegative real numbers, including zero. In particular, $(0,1)$-matrices are studied. On the basis of the classification of such matrices by type and of the general formula for the number of matrices of nullity $t$ valid for $t>n$ and $t\ge n>m$ (see [2]), an asymptotic (as $n\to\infty$) expansion is obtained for the total number of: (a) totally indecomposable matrices (Theorems 1 and 5), (b) partially decomposable matrices of given nullity $t\ge n$ (Theorems 2 and 4), (c) matrices with zero permanent (without using the inclusion-exclusion principle; Corollary of Theorem 2).