Problem 21 - Algebra and Trigonometry

Question

ID Numbers A company has 2844 employees. Each employee is to be given an ID number that consists of one letter followed by two digits. Is it possible to give each employee a different ID number using this scheme? Explain.

Step-by-Step Solution

Answer

No, because 2600 IDs are insufficient for 2844 employees.

Step 1: Calculate Total Possible IDs Per Letter

The ID number consists of one letter and two digits. Since there are 26 possible letters in the English alphabet, each letter allows for different combinations of two digits. Let's calculate the number of possible IDs for one letter. Each digit can be between 0 and 9, so there are 10 possibilities for the first digit and 10 for the second digit. This results in a total of \(10 \times 10 = 100\) different ID variations for each letter.

Step 2: Calculate Total Possible IDs

Now, multiply the number of possible ID combinations per letter by the number of letters. Since each letter has 100 combinations and there are 26 letters, the total number of unique IDs is \(26 \times 100 = 2600\).

Step 3: Determine if IDs Can Cover All Employees

There are 2844 employees who need unique IDs. Compare the total possible unique IDs (2600) with the number of employees (2844). Since 2600 is less than 2844, it's not possible to assign each employee a unique ID using this scheme.

Key Concepts

Permutations

In combinatorics, permutations refer to the arrangement of objects in a specific order. For the task of creating unique ID numbers, we consider permutations of letters and numbers to form unique sequences.

Each ID number comprises one letter followed by two digits, leading us to calculate permutations as follows:

• Selecting any of the 26 letters in the English alphabet forms the initial part of the ID number.
• Next, each of the two digit positions can be filled by any of the 10 digits (0-9), resulting in many possible combinations for each letter.
• Thus, computing total permutations per selected letter involves multiplying the possibilities for each digit, 10 for the first and 10 for the second, equating to 100 permutations per letter.

This setup allows permutations to illustrate how many unique ID arrangements can be formed initially.

Unique Identifiers

Unique identifiers are crucial in distinguishing between different elements within a set. In this situation, ID numbers serve as unique identifiers for employees.

The allocation of these identifiers follows a specific pattern:

• A single letter from the alphabet which can generate diverse possibilities for ID numbers.
• Two digits, each offering 10 unique numbers, making a robust identifier system.
• Using this strategy, there are initially 2600 unique identifiers, as there are 26 letters, each producing 100 unique number combinations.

The challenge arises when the number of employees (2844) exceeds the number of potential unique identifiers. Therefore, not every employee could receive a distinct ID number based on this scheme.

Mathematical Reasoning

Mathematical reasoning involves logical thinking to solve problems effectively. Here, it guides us in determining whether the ID scheme suffices for all employees.

The reasoning process unfolds in these steps:

• First, calculate the total number of possible IDs by considering the multiplying factor of choices—a concept known as the "rule of product" in combinatorics.
• We find that each letter with its 100 possible numeric combinations offers us 100 IDs.
• Multiplying 100 by the 26 letters yields 2600 potential IDs.

Since logical deduction finds 2600 IDs, reasoning further establishes that these are insufficient for 2844 employees. This discrepancy necessitates reevaluating the ID scheme or considering an additional identifier component (such as an extra digit) to cover all employees.