What is P($X_{1}>X_{2}>...X_{n-1}>X_{n}$)?

Question

Given $X_{i}$, $1 \leq i\leq n$, are independent random variables with uniform distributions on $[0, 1]$, what is P($X_{1}>X_{2}>...X_{n-1}>X_{n}$)?

I thought it would be something along the lines of $\int_{0}^1 \int_{x_{1}}^1 \int_{x_{2}}^1 ... \int_{x_{n-1}}^1 (1-x_{n})dx_{n}...dx_{3}dx_{2}dx_{1}$.

Answer 1

All possible orders are equally likely and there are $n!$ of them, so since the probability of any two variables being equal is zero the answer is $\frac{1}{n!}.$