Question

A series of articles about 26 athletic teams will appear over 26 consecutive weeks. How many ways can the articles be ordered? What if the first week's article must be about the current champion? What if team \(A\) and team \(B\) must be featured in consecutive weeks?

Short answer

1. \(26!\). 2. \(25!\). 3. \(25! \times 2\).

Step-by-step solution

Recognize the Initial Problem

To determine how many ways the articles can be ordered, we are essentially being asked to determine the number of permutations of 26 distinct items. This is because each article represents a unique team.

Calculate Permutations of 26 Teams

The number of permutations of 26 unique items (teams) is calculated using the factorial of 26, denoted as \(26!\). The formula is \(26! = 26 \times 25 \times 24 \times \ldots \times 1\).

Address the Condition for the Current Champion

If the first week's article must be about the current champion, we are fixing this position, leaving us 25 teams to arrange. Thus, the number of ways to order the remaining articles is \(25!\).

Consider the Condition for Teams A and B

If team A and team B must be featured in consecutive weeks, treat them as a single "block". This reduces the problem to ordering 25 items (24 teams + 1 block). Thus, the number of permutations is \(25!\). However, within the block, teams A and B can be ordered in 2 ways (AB or BA), leading to a total of \(25! \times 2\) ways.

Key concepts

Factorial

The concept of factorial is foundational in understanding permutations and arrangements.
A factorial, denoted by an exclamation mark (!), is the product of all positive integers up to a specified number.
For example, the factorial of 5, written as \(5!\), equals \(5 \times 4 \times 3 \times 2 \times 1 = 120\).
In combinatorial problems, factorials are used to compute the number of ways to arrange objects.

• The factorial function grows rapidly and is used when calculating arrangements of large sets, such as 26 items, which is the scenario with the athletic team articles.
• If you need the permutation of 26 teams, you compute \(26!\), illustrating the vast number of possible sequences for organizing the articles.

Learning to handle factorials is crucial when tackling more complex arrangement problems, as it allows for quick computation of permutations.

Combinatorial Problems

Combinatorial problems involve determining the number of ways to select and arrange items from a set. They can be particularly complex due to the variety of conditions and constraints involved. To solve these problems, we often use permutations and combinations. Permutations refer to arrangements where order matters, while combinations consider selections where order does not matter. In the exercise, since we're arranging articles about teams, permutations are relevant as each sequence produces a different result. For example:

• If no restrictions are imposed, determining the order of 26 teams requires computing the permutation \(26!\).
• If conditions are applied, such as fixing the position of the current champion or requiring two teams to be consecutive, these alter the count significantly.

Understanding these principles allows us to tailor our approach to each new scenario, ensuring we account for all possibilities correctly.

Arrangements

Arrangements are all about how we can sequence items. In this context, it means organizing the articles in various orders. When handling arrangements, particularly with additional conditions, it is important to break the problem down into manageable steps.For instance, if the first article must be about the current champion, you are setting a fixed point in the sequence, essentially reducing the problem to arranging the remaining 25 teams, hence \(25!\) arrangements.Another condition, such as keeping two teams (A and B) together, requires treating them as a single unit or "block." This block can be within any of the remaining spots, and within themselves, they can be ordered in two distinct ways (AB or BA). The calculation becomes \(25! \times 2\) as you first arrange the block among other teams and then flip A and B within the block.By focusing on the constraints and how they can lock or dictate the order of items, you make complex arrangement tasks more approachable.