In 2003, the 5th problem of the Chinese Mathematical Olympiad was as follows:
A company needs to hire a secretary, with a total of 10 people applying. The company manager decides to interview them in the order of their application, and the first 3 people will definitely not be hired. Starting from the 4th person, he will be compared with those who have been interviewed before. If his ability exceeds all those who have been interviewed before, he will be hired; otherwise, he will not be hired, and the next person will be interviewed. If none of the first 9 people are hired, then the last person interviewed will be hired.

Assuming the abilities of these 10 people are all different and can be ranked from 1st to 10th in terms of ability. Clearly, which person the company ends up hiring depends on the order in which these 10 people apply. There are 10! such permutations. We denote $A_{k}$ as the number of different application orders in which the person with the $k$-th ability is hired, and $\frac{A_{k}}{10!}$ as the probability of him being hired.