= Solution
The <independent and identically distributed random variables> have a <continuous probability distribution>, so any tie has probability zero. Their <joint probability density> is unchanged by a <permutation> of their indices: this is <order symmetry of independent random variables>. The $n!$ strict orderings are disjoint and exhaust the sample space apart from the zero-probability ties. Each ordering therefore has the same probability, and their probabilities sum to one.
\b[The probability is]
$$
\boxed{\mathbb P(X_1>X_2>\cdots>X_n)=\frac1{n!}.}
$$
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