91Ó°ÊÓ

In mathematics, permutations and combinations are two fundamental concepts in combinatorics that deal with arranging and selecting items. The key difference lies in whether the arrangement sequence matters or not.
In *permutations*, the order is important. For instance, if you are arranging the cities A, B, and C, the sequence

• ABC
• BCA
• CAB

represents different permutations because the order of cities is different in each case.
Conversely, in *combinations*, the order does not matter. This is the case in our exercise. It's about selecting groups, so combinations like

• ABC
• BCA
• CAB

are considered the same. Here, we only care about the "grouping" of cities and not the order they appear in. That's why we used the combinations formula to solve the problem.

The factorial of a non-negative integer n, denoted as \(n!\), is the product of all positive integers less than or equal to n. It plays a vital role in permutations and combinations.
The concept of factorial can be understood with simple examples:

• \(3! = 3 \times 2 \times 1 = 6\)
• \(4! = 4 \times 3 \times 2 \times 1 = 24\)
• \(5! = 5 \times 4 \times 3 \times 2 \times 1 = 120\)

Notably, by definition, \(0!\) is equal to 1. This might seem unusual, but it simplifies mathematical formulas and calculations. For example, in the combination formula \(C(n, r) = \frac{n!}{r! (n-r)!}\), using \(0! = 1\) allows us not to subtract any additional value when \(r = n\), as in our exercise solution.

Combinatorial mathematics is a field of mathematics focused on counting, arrangement, and combination of objects. It covers a wide array of practical applications, from basic counting to complex arrangement tasks.
In our context, combinatorial mathematics involves determining how many distinct routes exist for visiting a set of cities, where the sequence is not important.
Combinatorics covers several key areas:

• *Combinations and permutations*, where formulas are used to compute arrangements or selections.
• *Graph theory*, which can model networks like city routes, akin to our problem context.
• *Pigeonhole principle*, a simple yet powerful concept used in proving results about groupings.

The ultimate goal of combinatorial mathematics is to find ways to count and analyze these groupings efficiently, which can be invaluable in fields like computer science, logistics, and operational research.