Theorem

The probability $P_{1}$ of the realization of at least one among the events $A_{1},A_{2},\cdots,A_{M}$ is given by an alternating sum

$$P_{1}=S_{1}-S_{2}+S_{3}-S{4}+- \cdots \pm S_{M}.$$

Hence, the the probability that all cells are occupied is

$$p_{0}(N,M)=1-S_{1}+S_{2}-+\cdots=\sum_{\nu=0}^{M} (-1)^{\nu} \binom{M}{\nu}(1-\frac{\nu}{M})^{N}.$$

This probability is the solution to the recurrence formula that gave the number of arrangements with no cells empty divided by the total number of arrangements $M^N$.