Problem 17 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

To gain access to his account, a customer using an automatic teller machine (ATM) must enter a four-digit code. If repetition of the same four digits is not allowed (for example, 5555 ), how many possible combinations are there?

Step-by-Step Solution

Answer

The number of possible four-digit codes that do not have the same digit repeated is 9990.

Step 1: Calculate total number of four-digit codes.

To calculate the total number of possible four-digit codes, consider that there are 10 choices (digits from 0 to 9) for each of the 4 positions in the code. So, the total number of combinations is: \(10^4\)

Step 2: Calculate the number of unacceptable codes (same digit repeated).

For an unacceptable code (one with the same digit repeated), there are 10 options, one for each digit from 0 to 9.

Step 3: Subtract unacceptable codes from the total number of codes.

To find the number of acceptable four-digit codes, subtract the unacceptable codes from the total number of codes calculated in Step 1: Acceptable codes = Total codes - Unacceptable codes Acceptable codes = \(10^4 - 10\)

Step 4: Compute the final result.

Now, calculate the final number of acceptable codes: Acceptable codes = \(10^4 - 10 = 10000 - 10 = 9990\) There are 9990 possible four-digit codes that do not have the same digit repeated.

Key Concepts

Combinatorics

Combinatorics is a field of mathematics that deals with counting, arranging, and combination possibilities. It plays a crucial role in areas such as probability theory and computer science. In the context of the ATM code problem mentioned, combinatorics allows us to calculate the number of possible unique four-digit codes.

Imagine each digit in the four-digit code as a blank space that can be filled with any digit from 0 to 9. Initially, there are 10 options for each space. However, if we do not consider repetition of a single digit, the scenario changes. The combinatorial approach involves two stages: counting the total possibilities and then excluding the unwanted ones — the repeats. Calculating the total number of four-digit codes is just the beginning; combinatorics involves both generation and elimination methods to find the precise number of acceptable combinations.

• Inclusion: Every digit can be included in multiple ways alongside others.
• Exclusion: Removing combinations that break the rules (like repetitive sequences).

By combining inclusion and exclusion, we reach the solution for acceptable four-digit ATM codes.

Number Theory

Number theory is often referred to as the 'queen of mathematics' due to its foundational place in the discipline. It is concerned with the properties of numbers, particularly integers. In terms of our ATM code scenario, we're focused on a subset of number theory involving counting peculiar subsets of numbers.

For the ATM problem, the unacceptability of repetitive sequences such as '5555' represents a constraint rooted in number theory. It prevents any of the 10 single digits from repeating across all four positions. Here’s how number theory comes into play:

• We identify the total universe of numbers with four places, which is a power of 10 since we have 10 choices for each digit.
• We recognize a set of numbers (the repeated ones) that don't comply with our non-repetition rule.
• We perform an operation (subtraction) to exclude the non-compliant set from the total universe.

Through number theory, we understand the mathematical structure and constraints of our problem and effectively solve it.

Mathematical Reasoning

Mathematical reasoning encompasses the logical thinking and problem-solving processes used to understand and work with mathematics. This type of reasoning is not about computing; it's about comprehending concepts and applying logical steps to achieve conclusions. With regards to the ATM code problem, mathematical reasoning helps us break down the process into distinct, logical steps.

Each step of the solution stems from a rational decision:

• Deciding how to calculate the total number of possibilities.
• Determining which codes are unacceptable under the given rules.
• Utilizing arithmetic operations (like subtraction) to refine our count.

Mathematical reasoning also involves error-checking and verification. In our case, we ensure that the final count of possible four-digit codes doesn't include any repetitions. This methodology ensures that our solution is not only calculated correctly but also aligns with the real-world application of such codes.