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Chapter 7: Problem 42

In how many ways can five people boarding a bus be seated if the bus has eight vacant seats?

Short Answer

Expert verified
There are 6720 different ways in which the five people can be seated on a bus with eight vacant seats. We calculated this using the concept of permutations and the formula \(\frac{n!}{(n - r)!}\), where \(n\) is the total number of seats (8) and \(r\) is the number of people (5).

Step by step solution

01

Identify the number of choices and decision points

We have 5 people who need to be seated in 8 vacant seats. This means that we have a decision to make for each person for which seat they will sit in. For the first person, there are 8 choices, for the second person 7, for the third 6, and so on.
02

Calculate the number of permutations

In order to find the number of seating arrangements we can make, we will have to calculate the number of permutations for 5 people and 8 seats. We can do this using the formula: Number of permutations = \(\frac{n!}{(n - r)!}\) where 'n' is the total number of seats (8 in this case) and 'r' is the number of people (5 in this case).
03

Substitute the values in the formula

Substituting the values in the permutation formula, we get: Number of permutations = \(\frac{8!}{(8-5)!}\)
04

Perform the calculations

Now, let's simplify this expression by calculating the factorials: Number of permutations = \(\frac{8\times7\times6\times5\times4\times3\times2\times1}{3\times2\times1}\) Number of permutations = \(\frac{40320}{6}\)
05

Calculate the final result

Finally, let's calculate the result by dividing 40320 by 6: Number of permutations = 6720 So, the five people can be seated in 6720 different ways on a bus with eight vacant seats.

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Key Concepts

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

Combinatorics
Combinatorics is a branch of mathematics dealing with counting, combination, and permutation of elements in sets. It's widely used in problems where you have to determine the number of ways something can occur. In this exercise, combinatorics helps us figure out how five people can be seated in eight vacant seats.

By understanding the choices available (8 seats), and recognizing the decision for each person (which seat to occupy), we use combinatorial principles to solve the problem. Permutations are a key part of combinatorics, where the order of arrangement matters, such as in seating people.

Combinatorial topics often involve:
  • Permutations: Arranging objects where order matters.
  • Combinations: Selecting objects where order doesn鈥檛 matter.
  • Graph theory and network analysis.
Knowing when and how to apply these is crucial for solving many real-world and theoretical problems.
Factorials
Factorials are foundational in calculating permutations and combinations. The factorial of a number (n!) is the product of all positive integers less than or equal to that number. For instance, 5! = 5 脳 4 脳 3 脳 2 脳 1 = 120.

Factorials allow us to simplify expressions involving permutations. In our exercise, we calculate the number of ways to seat five people using the permutation formula: \[ \frac{8!}{(8-5)!} \]. This involves computing 8! and (8-5)!, simplifying the expression to find the number of arrangements.

Factorials are particularly useful in:
  • Analyzing arrangements and orders.
  • Determining combinations of items.
  • Simplifying complex mathematical expressions.
Understanding how factorials work helps unlock many problems in discrete mathematics.
Discrete Mathematics
Discrete Mathematics deals with objects that can be counted in a distinct, separated manner. Unlike continuous systems, discrete mathematics focuses on countable, often finite sets.

In our bus seating problem, discrete mathematics helps break down the steps to count the exact number of seating arrangements for people in seats. It involves concrete structures like:
  • Integers and graphs.
  • Logical statements and set theory.
  • Combinatorics and probability.
This branch underpins computer science, cryptography, and network algorithms, making it essential for modern technology. Learning discrete mathematics provides tools to approach and solve logical problems systematically.

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