Problem 35 - Algebra and Trigonometry with Analytic Geometry

Question

ATM access code A customer remembers that \(2,4,7\), and 9 are the digits of a four-digit access code for an automatic bank-teller machine. Unfortunately, the customer has forgotten the order of the digits. Find the largest possible number of trials necessary to obtain the correct code.

Step-by-Step Solution

Answer

The largest possible number of trials is 24.

Step 1: Identify the Problem

We need to find the largest number of trials required to determine the correct order of four distinct digits (2, 4, 7, 9) to form an access code.

Step 2: Permutation Formula

The order in which the digits are arranged is important, so we use the formula for permutations. For a set of 4 distinct digits, the total permutations is calculated by \(n!\), where \(n\) is the number of digits.

Step 3: Calculate the Permutations

There are 4 digits: 2, 4, 7, and 9. Thus, the number of permutations is \(4!\). Calculate \(4! = 4 \times 3 \times 2 \times 1 = 24\).

Step 4: Conclusion

The largest possible number of trials necessary to obtain the correct code is the number of different permutations of the 4 digits.

Key Concepts

Factorial

In mathematics, the term 'factorial' is an important concept used to find the total number of ways things can be arranged. The factorial of a number, denoted by an exclamation mark (!), is the product of all positive integers less than or equal to that number. For example, the factorial of 4, written as \(4!\), is calculated as 4 multiplied by 3, followed by 2, and then 1. This results in \(4! = 4 \times 3 \times 2 \times 1 = 24\).

Factorials are commonly used in statistics, mathematics, and particularly in solving permutation and combination problems. They help in calculating how many different ways you can arrange a set of items. The factorial begins with the highest number and counts down to 1, multiplying each number as you go on.

Whenever you are determining the number of ways to order a group of items where each item is unique and has a specific position in the sequence, you rely on factorial to give you the answer.

Combinatorics

Combinatorics is a branch of mathematics dealing with combinations of items and the counting of these combinations. It's quite useful when solving problems related to permutations and combinations, where the order or selection of items matters.

Combinatorics explores various ways we can arrange or group items, whether we are looking at the order or just groups itself. For example, in permutations, we're interested in how many ways we can arrange a set number of items where the order is important. On the other hand, combinations are about selecting items from a larger pool, where the order does not matter.

This field of study provides essential tools for solving problems involving arrangements, selections, and listings, and is fundamental in understanding probabilities and statistical distributions. It's a fascinating subject that can be simple or complex based on the problem at hand, but is always about the art of choice and arrangement.

Arrangements

The concept of arrangements refers to how items can be ordered or positioned. In the context of permutations, the arrangement is critical because it determines the sequence or order of items.

In the problem of finding the ATM access code, we deal with the arrangement of four digits, 2, 4, 7, and 9. Here, every permutation represents a different arrangement, or sequence, of these digits. Since these digits must be placed in a specific order to form the correct access code, the arrangement is what we're calculating through permutations.

Arrangements become necessary in any situation where the sequence or specific order of items yields different results, such as passwords, seating arrangements, or schedule planning. Understanding how to compute and interpret these arrangements allows you to solve real-world problems effectively.