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Chapter 8: Problem 13

How many derangements are there of a set with seven elements?

Short Answer

Expert verified
There are approximately 1854 derangements of a set with seven elements.

Step by step solution

01

- Understand the Problem

A derangement is a permutation of a set where none of the elements appear in their original positions. The task is to find the number of derangements for a set with seven elements.
02

- Use the Derangement Formula

The number of derangements of a set with n elements is given by the formula: \[ !n = n! \times \bigg( \frac{(-1)^0}{0!} + \frac{(-1)^1}{1!} + \frac{(-1)^2}{2!} + ... + \frac{(-1)^n}{n!} \bigg) \]
03

- Apply the Formula for 7 Elements

Substitute 7 for n in the derangement formula: \[ !7 = 7! \times \bigg( \frac{(-1)^0}{0!} + \frac{(-1)^1}{1!} + \frac{(-1)^2}{2!} + \frac{(-1)^3}{3!} + \frac{(-1)^4}{4!} + \frac{(-1)^5}{5!} + \frac{(-1)^6}{6!} + \frac{(-1)^7}{7!} \bigg) \]
04

- Calculate the Factorials and Signs

Calculate each term separately: \[ !7 = 5040 \times \bigg( 1 - 1 + \frac{1}{2!} - \frac{1}{3!} + \frac{1}{4!} - \frac{1}{5!} + \frac{1}{6!} - \frac{1}{7!} \bigg) \]Simplify the expression within the parentheses.
05

- Simplify the Summation

Combine the computed fractions: \[ 1 - 1 + \frac{1}{2!} - \frac{1}{3!} + \frac{1}{4!} - \frac{1}{5!} + \frac{1}{6!} - \frac{1}{7!} = 0 + 0.5 - 0.1667 + 0.0417 - 0.0083 + 0.0014 - 0.0002 \]Summing these gives approximately 0.3679.
06

- Calculate the Final Result

Multiply the result by 5040: \[ !7 \times 0.3679 \times 5040 \times 0.3679 \]So, \[ !7 \times = 1854 \] (approximated to the nearest whole number).

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Key Concepts

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

Permutations
Permutations play a critical role in various mathematical concepts, including derangements. In simple terms, a permutation is an arrangement of all the members of a set into a sequence or order. For instance, suppose we have a set with three elements {A, B, C}. The possible permutations would be ABC, ACB, BAC, BCA, CAB, and CBA.
Permutations help find the number of possible ways to arrange elements, and this concept extends to derangements, where we specifically look at arrangements that meet certain conditions.
A key formula to remember is the total number of permutations of a set with n elements is given by the factorial of n, denoted as n!. This means for a set with three elements, the number of permutations would be:
  • 3! = 3 脳 2 脳 1 = 6
Factorials
Factorials are fundamental to understanding combing and organizing sets of elements. They are denoted with an exclamation mark (n!) and represent the product of all positive integers up to that number. For example:
  • 5! = 5 脳 4 脳 3 脳 2 脳 1 = 120
  • 7! = 7 脳 6 脳 5 脳 4 脳 3 脳 2 脳 1 = 5040
Factorials are especially useful in problems involving permutations and combinations. In the context of derangements, factorials help us determine the total number of arrangements before applying additional constraints.
Factorials also appear in the derangement formula, which extends factorial functions with alternating signs for more extensive calculations.
A good practice is to get comfortable calculating factorials as they frequently appear in combinatorial mathematics.
Combinatorics
Combinatorics is a branch of mathematics dealing with combinations, permutations, and counting principles. It provides tools to solve problems related to arranging, grouping, and selecting objects.
Topics in combinatorics include:
  • Permutations
  • Combinations
  • Derangements
Understanding combinatorics helps break down complex arrangements into simpler components. For instance, derangements are a specialized type of permutation where no element is in its original position.
The derangement formula:ewline \[ !n = n! \times \bigg( \frac{(-1)^0}{0!} + \frac{(-1)^1}{1!} + \frac{(-1)^2}{2!} + \ldots + \frac{(-1)^n}{n!} \bigg) \]ewlineuses combinatorial principles to provide the count of such arrangements.
Combinatorics is a powerful field, enabling students to solve real-world problems involving complex arrangements.

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