Arrange n distinct books in a row, of which m are volumes of one science series (m < n). What is the probability that those m volumes appear in ascending volume order (not necessarily consecutively)?

Introduction / Context:
We randomly arrange n distinct books, of which m belong to a numbered series. We only care about the relative order among the m volumes; they need not be adjacent. What is the chance their left-to-right order is ascending by volume number?

Given Data / Assumptions:

• All n! permutations are equally likely.
• The m special volumes are distinct and labeled (by volume numbers).
• We examine their relative order induced by any permutation.

Concept / Approach:
Among the m books of the series, all m! relative orders are equally likely when one looks only at their induced order in any full permutation. Exactly one of these m! orders is ascending.

Step-by-Step Solution:
Number of possible relative orders for the m volumes = m!.Favorable relative orders (ascending) = 1.Probability = 1 / m!.

Verification / Alternative check:
This is a standard symmetry argument: conditioning on the positions occupied by the m volumes, all m! permutations of those m among the chosen positions are equally likely.

Why Other Options Are Wrong:
m!/n! involves absolute positions, not just relative order; 1/(n−m)! is unrelated; 1/n! is far too small.

Common Pitfalls:
Assuming the volumes must be consecutive (not required). Only relative increasing order matters.

Final Answer:
1/m!