Problem of the Day #77

POTD February 23, 2024
Author’s note: we let awesomeming327 cook.

Ten people pick real numbers. The n^{\text{th}} person picks at random from the interval [0, n]. If the probability that the sequence of numbers picked by the 1^{\text{st}}, 2^{\text{nd}}, \ldots, 10^{\text{th}} person is strictly increasing can be denoted by \omega. Compute \sqrt[9]{(10!)^2\cdot\omega}.