Problem 46 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use the letters \(\mathbf{A}, \mathbf{B}, \mathbf{C},\) and \(\mathbf{D}\). Write all permutations of the letters when the letters \(\mathrm{B}\) and \(\mathrm{C}\) must remain between the letters \(\mathrm{A}\) and \(\mathrm{D}\).

Step-by-Step Solution

Answer

The total number of permutations where B and C remain together and between A and D is 12.

Step 1: Understand the Constraint

Recognize that the letters B and C must be together and be placed between A and D. This forms essentially three groups: A, BC, D.

Step 2: Permute the three groups

Determine all possible placements of the three groups. The total number of permutations of \(n\) groups is \(n!\). In this case, we have 3 groups, so there are \(3! = 3 \times 2 \times 1 = 6\) possible placements.

Step 3: Permute within the BC group

For each placement of the three groups, there are also two possible placements for B and C, since they can swap places. So, for each placement calculated in step 2, the letters B and C can be arranged in \(2!\) ways.

Step 4: Calculate the Total Permutations

Multiply the number of placements of the three groups by the number of placements of B and C within their group. Therefore, the total number of permutations is \(3! \times 2! = 6 \times 2 = 12\)

Key Concepts

Combinatorics

Combinatorics is the branch of mathematics focused on counting, grouping, and arranging objects. It helps us figure out how many different arrangements are possible in specific situations. In this exercise, we explore how to use combinatorics to arrange the letters A, B, C, and D with a specific condition.
The problem involves not only counting the total arrangements but also satisfying a specific constraint, which is where combinatorics shines. By understanding how to break down the arrangement into manageable steps, we can solve it efficiently.
Think of combinatorics as a toolbox. When faced with arranging objects, you select the appropriate tools like permutation or combination to find the solution efficiently.

Factorial

Factorials are key in permutations. The factorial of a non-negative integer n, written as \( n! \), is the product of all positive integers less than or equal to n. For example, \( 3! = 3 \times 2 \times 1 = 6 \).
In the context of permutations, factorials tell us the number of different ways to order a set of items. For example, if you have three items, there are \( 3! \) ways to arrange them. This explains why, in our exercise, there are 6 unique ways to arrange the groups A, BC, and D.
By mastering factorials, students unlock the potential to solve a variety of combinatorial problems, making this concept fundamental in any study of permutations.

Constraint Handling

When handling constraints in permutation problems, it is crucial to recognize these restrictions and modify the solution accordingly. In our problem, B and C must remain between A and D. This means a direct permutation of all four letters is not possible.
Instead, we treat B and C as one unit 'BC,' effectively transforming our problem into a permutation involving three groups: A, BC, and D. Once these groups are permuted, we consider the arrangements within the 'BC' group, ensuring all constraints are respected.
This approach highlights how attention to constraints helps narrow down possible permutations, making problems easier to solve without overlooking crucial conditions.

Grouping

Grouping is an essential strategy in permutations when constraints or special conditions are present. By grouping objects together, you simplify the permutation problem, focusing on their combined position rather than individual spots.
In this exercise, the letters B and C form one group due to the constraint. This effectively reduces the number of items from four to three, making it easier to calculate arrangements. By considering this 'BC' group as a single unit in the counting process, solving the permutation can focus on fewer entities.
Grouping is not only useful in mathematics but can apply to any situation requiring organized thinking or planning, underscoring its utility beyond this exercise.