Combinatorics: interview strategy problem (secretary problem)

The problem is to answer some questions about the following hiring strategy:

• Reject the first r-1
• Accept the first one after r-1 with ranking better than the first r-1

Note that there is an absolute rank of all n applicants so there is a best applicant i.e. one with the best rank, but is unknown. We only know the relative ranks of the interviewed applicants.

We define $P_n(r)$ as the probability of accepting the absolute best ranked using the above strategy.

For example $P_4(2)$ if I am not wrong is 11/24. If we look at all the permutations of the absolute rank of all the applicants (4!).

1*** (6) best applicant is already rejected here

2*** (6) best applicant will definitely be taken here because nobody is better than the second best than the first best

First I have to show that $P_(3) = \frac (\frac + \frac + . + \frac )$ , derive the probability of accepting the applicant with the absolute rank 1 among n applicants at the k-th interview then find a recursive formula of $P_n(r)$ in the form $P_n(r)=A+BP_n(r+1)$ .

I have done some research on the problem and it seems it has a classical name "Secretary Problem".