Problem 19 - Algebra and Trigonometry

Question

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, it is not possible; 2600 IDs are insufficient for 2844 employees.

Step 1: Calculate Total Letter Combinations

The ID number starts with a letter. Since there are 26 letters in the English alphabet, there are 26 possible choices for the letter.

Step 2: Calculate Total Digit Combinations

The ID follows the letter with two digits. There are 10 possible digits (0 through 9) for each digit place. Therefore, the two digits can be combined in \(10 \times 10 = 100\) different ways.

Step 3: Calculate Total ID Combinations

To find the total number of unique ID numbers, multiply the total number of letter combinations by the total number of digit combinations: \(26 \times 100 = 2600\).

Step 4: Compare Total ID Combinations to Number of Employees

With the calculated 2600 possible unique IDs and 2844 employees, there are not enough unique IDs available since 2600 < 2844.

Key Concepts

Unique Identifiers

In today's world, providing each element or individual with a distinct label or code is crucial. This is what we call a "unique identifier." In the context of the exercise, the unique identifier is the ID number given to each employee. The uniqueness is essential to avoid confusion and ensure clear identification.

• Each identifier must be different to ensure that it corresponds to one and only one employee.
• Unique identifiers help in efficiently managing and processing data.

Think about the practical applications, like how a college assigns student IDs to keep track of student records. Without unique identifiers, mixing up information would be inevitable. In this exercise, letters and numbers are combined to form IDs. However, we must ensure that the combination is sufficient to cover every unique entity, which, in this case, are the employees.

Permutations in Mathematics

Permutations refer to arrangements or sequences in which items can be ordered or arranged. In the exercise's scenario, permutations are utilized to determine the possible combinations of letters and numbers for generating unique identifiers.

• A permutation takes into account the order in which items appear.
• The more elements (e.g., digits, letters) you can combine, the greater the number of possible permutations.

For instance, if we are arranging two different digits, say 3 and 5, they can appear as 35 or 53, creating two distinct permutations. In the exercise, the permutations involve one letter followed by two digits, calculated as \(26 \times 100 = 2600\). Though this generates numerous unique combinations, unfortunately, it's not enough to assign a different ID to all 2844 employees.

Mathematical Problem Solving

Approaching a mathematical problem often involves breaking it down into smaller, more manageable steps. This is clearly illustrated in the exercise, which asks whether enough unique ID combinations can be made for all employees. Here's how problem-solving in this context works:

• Identify what you need to find – the total number of unique IDs necessary.
• Break down the components, like the option of letters and digits, to calculate possible permutations separately.
• Compare the number of permutations with the actual need (number of employees).

Solving the problem means deducing whether the combinations suffice or if the scheme needs reevaluation. The methodical approach helps in reaching conclusions effectively, showcasing why understanding the process is vital in mathematical problem solving.