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Chapter 6: Problem 33

$$ \text { Write all the } 3 \text {-permutations of }\\{s, t, u, v\\} \text {. } $$

Short Answer

Expert verified
The 24 3-permutations of the set {s, t, u, v} are: stu, stv, suv, svt, svt, sut, tsu, tsv, tus, tvs, tvu, tuv, ust, usv, uts, utv, uvt, uvs, vst, vsu, vts, vtu, vus, and vut.

Step by step solution

01

Understanding permutation

A permutation is an arrangement of objects in a specific order. In this case, we want to find the arrangements of three elements from the given set {s, t, u, v}. There are 4 elements in the set, and we want to arrange them 3 at a time. This can be represented as P(4, 3).
02

Calculate the number of 3-permutations

To calculate the number of 3-permutations, we can use the formula: \(P(n, r) = \frac{n!}{(n-r)!}\) Here, n = 4 (total number of elements) and r = 3 (number of elements to arrange). Plugging in the values, we get: \(P(4, 3) = \frac{4!}{(4-3)!} = \frac{4!}{1!} = \frac{4 脳 3 脳 2 脳 1}{1} = 24\) So, there are 24 possible 3-permutations of the set {s, t, u, v}.
03

List all the 3-permutations

Now, let's list all the 3-permutations. We can do this by fixing one element at a time and listing the permutations of the other two elements. 1. Fix s first: stu, stv, suv, svt, svt, sut 2. Fix t first: tsu, tsv, tus, tvs, tvu, tuv 3. Fix u first: ust, usv, uts, utv, uvt, uvs 4. Fix v first: vst, vsu, vts, vtu, vus, vut In total, we have found 24 3-permutations, which matches the result calculated in step 2.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Arrangement of Objects
Understanding how to arrange objects is a fundamental concept in permutations. In a permutation, the order of the objects matters. For instance, arranging the letters {s, t, u} is different from arranging them as {t, s, u}. When we talk about arrangements, we're organizing these objects in a sequence where each sequence is unique due to the order of elements.
Consider the task of arranging three letters from a set of four: {s, t, u, v}. Here, we choose any three letters and organize them in every possible order. This way, the specific order creates different arrangements.
Permutation Formula
The permutation formula plays a crucial role in calculating how we can arrange objects. It is represented as:
\(P(n, r) = \frac{n!}{(n-r)!} \)
In this formula, \(n\) is the total number of objects available, and \(r\) is the number of objects we want to arrange. The symbol \(!\) denotes a factorial, meaning a series of multiplying descending numbers.
  • For \(n = 4\) and \(r = 3\), our calculation becomes \(P(4, 3) = \frac{4!}{1!} = 24\).
  • Here \(4! = 4 \times 3 \times 2 \times 1\), representing all ways to arrange four items, and \(1!\) is just 1.
This result confirms there are 24 ways to arrange three letters chosen from the set {s, t, u, v}.
Combinatorics
Combinatorics deals with counting, arranging, and understanding the deeper properties of sets of objects. Permutations are a part of combinatorics.
When dealing with problems like arranging 3 elements from a set of 4, combinatorics helps us understand not just the quantity but also the logical structure of these arrangements.
  • It organizes problems into solvable steps using tools like the permutation formula.
  • By understanding combinatorics, solving arrangement problems becomes more systematic as it provides a framework to count and arrange items without errors.
With this knowledge, breaking down complex arrangement tasks into smaller, manageable parts becomes feasible, enabling efficient solutions.

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