Question

A national organization plans to issue its members a 4 -character ID code. The first character can be any letter other than O. The last 3 characters are to be 3 different digits. If the organization has 25300 members, will they be able to assign each member a different ID code? Explain.

Short answer

No, they will not be able to assign a different ID code to each member, as there are only 18,000 possible codes.

Step-by-step solution

Determine the number of options for the first character

The first character can be any letter except 'O'. Since there are 26 letters in the alphabet, excluding 'O' leaves us with 25 possible choices for the first character.

Determine the number of options for the last three characters

The last three characters are supposed to be three different digits. For the first digit, there are 10 possible choices (0-9). For the second digit, there are 9 remaining choices (since it must be different from the first digit). For the third digit, there are 8 remaining choices (since it must be different from the first two digits).

Calculate the total number of possible ID codes

To find the total number of possible ID codes, multiply the number of options for each part of the ID code: different letters for the first character = 25different digits for the second character = 10different digits for the second character = 9different digits for the second character = 8Hence, the total number of ID codes is given by: \[ 25 \times 10 \times 9 \times 8 = 18000 \]

Compare the number of possible ID codes to the number of members

The total number of different ID codes is 18,000. Since the organization has 25,300 members, which is greater than 18,000, they will not be able to assign a different ID code to each member.

Key concepts

Permutations and Combinations

When solving problems related to arranging or grouping items, it's essential to understand permutations and combinations.
Permutations are arrangements where order matters. For example, arranging three books on a shelf.
Combinations are selections where order doesn't matter. An example is selecting three fruits from a basket of different fruits.

Let's break it down with a focus on the national organization's ID code.

In the exercise, both permutations and combinations play a role, but primarily permutations.
Why? Because the order of the digits matters (123 is different from 321).
The first letter can be any of 25 possible letters.
But for the last three digits, order is crucial.
We need different digits in each position.

As a result, understanding permutations enables us to solve these kinds of ID generation problems.

Problem-Solving Steps

Solving a combinatorial problem involves logical steps.
Let's see the strategy applied here:

• Identify the constraints: Here, the first character can't be 'O', and the other three must be different digits.
• Calculate possible choices: For the first character, we have 25 options (all letters minus 'O'). For the digits, the choices differ because they must be unique.
• Multiply the choices: For each step, multiply the number of available options. For digits, it's important they are different.
• Solution: We calculated in the problem as follows: \(25 \times 10 \times 9 \times 8 \).

The final step is comparing the result to the requirement (25300 members). Since 18000 is less than 25300, we concluded there's an issue.

This methodical approach ensures no detail is missed and leads us to accurate results.

ID Code Generation

Generating unique IDs is a common requirement in many systems.
It helps keep things organized and prevents confusion.

In our exercise, the objective was to create distinct 4-character ID codes.
Each part of the code had specific rules.
These were:

• The first character from the alphabet, excluding 'O'.
• The next three characters had to be unique digits (0-9).

Each step must ensure all conditions are met.
By following a combinatorial approach, we determined the total possible codes.
25 choices for the letter, and then unique 10, 9, and finally 8 choices for the digits.
Even though we had significant numbers, it wasn't enough for 25300 members.

This exercise demonstrates the importance of efficient ID code generation and the need for careful combinatorial calculations.