Problem 50 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Seven people arrive at the ticket counter of a cinema at the same time. In how many ways can they line up to purchase their tickets?

Step-by-Step Solution

Answer

The seven people can line up to purchase their tickets at the cinema in 5040 different ways.

Step 1: Understanding Factorials

A factorial is represented by an exclamation mark and is the product of all positive integers less than or equal to the number. For example, 5! = 5 * 4 * 3 * 2 * 1 = 120. In this exercise, we have 7 people, so we need to find the value of 7!.

Step 2: Calculating 7!

Now, we need to find the product of all positive integers less than or equal to 7. 7! = 7 * 6 * 5 * 4 * 3 * 2 * 1

Step 3: Perform the Multiplication

Let's multiply these integers together: 7! = \(7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 5040\)

Step 4: Interpret the Result

There are 5040 different ways that the seven people can line up to purchase their tickets at the cinema.

Key Concepts

Factorials

Factorials are a fundamental concept in mathematics, primarily used in permutations and combinations. When you see a number followed by an exclamation mark, such as "7!", it denotes a factorial. Calculating a factorial is straightforward: multiply that number by every positive integer below it down to 1.
For instance, the factorial of 7, expressed as "7!", means:

• Start with 7
• Multiply by 6
• Then multiply by 5
• Continue multiplying until you reach 1

This gives us: \[ 7! = 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 5040 \]Factorials grow rapidly with larger numbers, quickly leading to large results. This growth is why they are so useful in counting arrangements or permutations.

Combinatorics

Combinatorics is a branch of mathematics concerning the study of finite or countable discrete structures. It plays a crucial role in various fields such as computer science, physics, and statistics. At its core, combinatorics revolves around counting the number of ways things can be arranged or combined.
In our cinema ticket counter example, we are interested in permutations. Permutations are specific arrangements of a set of objects. When calculating permutations, order matters. In this scenario, we are trying to determine how 7 people can stand in line, each in a unique position and in different orders.
This is where factorials come into play. The calculation of permutations of 7 individuals is represented by 7!, which ultimately results in 5040 unique ordering possibilities. Understanding permutations helps us solve problems where the arrangement or sequence is a factor, like seating arrangements or schedules.

Mathematical Problem Solving

Mathematical problem solving is a key skill that enables tackling a wide range of problems using logical reasoning and mathematical concepts. The key is to break down a problem into manageable steps, which often simplifies the process.
In the cinema problem, the first step involved understanding what was being asked: finding the number of different ways 7 people can line up. Recognizing this as a permutation problem helped us identify the need for a factorial calculation. Each step was logically defined:

• Identify the problem type and relevant mathematical concept (permutations & factorials)
• Translate the problem into a mathematical expression (7!)
• Carry out the calculations
• Interpret the results to offer a solution

Practicing these problem-solving techniques allows learners to approach unfamiliar problems with confidence and structure, often making complex problems feel manageable and solvable.