Problem 17 - Calculus for Biology and Medicine

Question

Six customers arrive at a bank at the same time. Only one customer at a time can be served. In how many ways can the six customers be served?

Step-by-Step Solution

Answer

There are 720 ways to serve the six customers.

Step 1: Understand the Problem

We need to determine in how many different ways six customers can be served one after the other.

Step 2: Recognize Permutation Scenario

Serving the customers one at a time is a permutation problem since the order in which customers are served matters.

Step 3: Use the Permutation Formula

The number of ways to arrange 6 customers in line is found using the factorial function, denoted as \(6!\).

Step 4: Calculate the Factorial

The factorial of a number \(n\) is the product of all positive integers less than or equal to \(n\). So, \(6! = 6 \times 5 \times 4 \times 3 \times 2 \times 1\).

Step 5: Perform the Calculation

Calculate \(6!\) as follows: \(6 \times 5 = 30, 30 \times 4 = 120, 120 \times 3 = 360, 360 \times 2 = 720, 720 \times 1 = 720\).

Step 6: Verify the Solution

The calculated \(6!\) is 720, which means there are 720 different ways to serve the six customers.

Key Concepts

Understanding Factorials

A factorial, denoted by an exclamation mark (!), is a mathematical operation applied primarily in permutations and combinations, such as our bank customers' example. To compute the factorial of a number such as 6, we multiply the number by every positive integer less than it. For instance, calculating the factorial of 6, written as \(6!\), involves the operation \(6 \times 5 \times 4 \times 3 \times 2 \times 1\).

This results in \(6! = 720\), the total number of ways to arrange the six customers in line.
Factorials grow extremely fast as numbers increase, which makes them incredibly useful for solving permutation and combination problems as seen here.

• Factorials help in determining the number of ways to arrange or order a set of objects.
• They are crucial in permutations where order matters, like arranging customers in line.

Being able to compute factorials efficiently can greatly simplify solving similar problems.

Basics of Combinatorics

Combinatorics is a branch of mathematics focused on counting, arrangement, and combination of objects. It's highly relevant in solving real-world problems like determining how to serve customers efficiently.
In the context of the bank problem, combinatorics helps identify possible arrangements. Even though combinatorics broadly covers combinations and permutations, the emphasis is on permutations when order matters.

With permutations, each unique order of serving the customers counts, unlike combinations which are concerned with selecting objects without regard for order. Combinatorics provides tools for counting arrangements when order is a key factor, like in the bank scenario.

• Combinatorics helps identify patterns and optimize processes like queue management.
• It's applicable in scheduling, logistics, and any problem involving arrangements.

Understanding combinatorics can improve strategic planning and decision-making in everyday situations.

Solving Permutation Problems

Permutation problems are about finding the number of ways to arrange a set of items where the order is significant. In the bank customer scenario, we want to know how to line up six customers in every possible way. Since order is crucial, permutations apply rather than combinations.

The solution involves finding the factorial of the number of objects (customers). In our example, this is \(6!\). Permutations directly use factorials to calculate all possible orders, unlike combinations where order isn't considered.
To summarize the steps to solve permutation problems:

• Identify if the problem involves arrangement where order matters.
• Use the factorial formula to compute the result.

Such problems often arise in scheduling, arranging seats, and determining jobs for a queue.