  The Secretary problem is a classical sequential decision-making question that
can be succinctly described as follows: a set of rank-ordered applicants are
interviewed sequentially for a single position. Once an applicant is
interviewed, an immediate and irrevocable decision is made if the person is to
be offered the job or not and only applicants observed so far can be used in
the decision process. The problem of interest is to identify the stopping rule
that maximizes the probability of hiring the highest-ranked applicant. A
multiple-choice version of the Secretary problem, known as the Dowry problem,
assumes that one is given a fixed integer budget for the total number of
selections allowed to choose the best applicant. It has been solved using tools
from dynamic programming and optimal stopping theory. We provide the first
combinatorial proof for a related new \emph{query-based model} for which we are
allowed to solicit the response of an expert to determine if an applicant is
optimal. Since the selection criteria differ from those of the Dowry problem we
obtain nonidentical expected stopping times.
  Our result indicates that an optimal strategy is the $(a_s, a_{s-1}, \ldots,
a_1)$-strategy, i.e., for the $i^{th}$ selection, where $1 \le i \le s$ and $1
\le j = s+1-i \le s$, we reject the first $a_j$ applicants, wait until the
decision of the $(i-1)^{th}$ selection (if $i \ge 2$), and then accept the next
applicant whose qualification is better than all previously appeared
applicants. Furthermore, our optimal strategy is right-hand based, i.e., the
optimal strategies for two models with $s_1$ and $s_2$ selections in total
($s_1 < s_2$) share the same sequence $a_1, a_2, \ldots, a_{s_1}$ when it is
viewed from the right. When the total number of applicants tends to infinity,
our result agrees with the thresholds obtained by Gilbert and Mosteller.
