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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

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

Hence, the the probability that all cells are occupied is

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

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$.

Tiziano Mengotti 2004-03-27