Systems of distinct representatives for random sets

In this work estimates from below are obtained for the probability that the permanent of a random $n\times m$ $(0,1)$-matrix is positive. Using this estimate, it is shown that a random collection of subsets $X_1,\dots,X_n$ of the set $X$ of $m$ elements as $m\to\infty$ has a system of distinct representatives with probability close to one.
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