Digits and Bases · Superchallenge problem

Every positive integer $n$ can be written uniquely in the factorial base as $n = d_L \cdot L! + d_{L-1}\cdot (L-1)! + \cdots + d_2 \cdot 2! + d_1 \cdot 1!$ with digits $0 \le d_k \le k$ and leading digit $d_L \ne 0$. For example $17 = 2\cdot 3! + 2 \cdot 2! + 1 \cdot 1!$. How many integers $n$ with $1 \le n \le 719$ have NO digit equal to $0$ in their factorial-base representation?