  In this paper, we find explicit formulas or generating functions for the
cardinalities of the sets $S_n(T,\tau)$ of all permutations in $S_n$ that avoid
a pattern $\tau\in S_k$ and a set $T$, $|T|\geq 2$, of patterns from $S_3$. The
main body of the paper is divided into three sections corresponding to the
cases $|T|=2,3$ and $|T|\geq 4$. As an example, in the fifth section, we obtain
the complete classification of all cardinalities of the sets $S_n(T,\tau)$ for
$k=4$.
