  We will show that for $q<p$ there exists an $\al < \infty$ such that \[
\pi_{pq}(T) \pl \le c_{pq} \pi_{pq}^{[n^{\alpha}]}(T) \mbox{for all $T$ of rank
$n$.}\] Such a polynomial number is only possible if $q=2$ or $q<p$.
Furthermore, the growth rate is linear if $q=2$ or
$\frac{1}{q}-\frac{1}{p}>\frac{1}{2}$. Unless
$\frac{1}{q}-\frac{1}{p}=\frac{1}{2}$ this is also a necessary condition .
