A company plans to assign identification numbers to its employees. Each number is to consist of four different digits from...

1. Translate the problem requirements: We need to create 4-digit identification numbers where all digits are different, digits come from 0-9, and the first digit cannot be 0. We need to count total possible arrangements.
2. Analyze position-by-position constraints: Determine how many digit choices are available for each of the four positions, considering the restriction that digits must be different and first digit cannot be 0.
3. Apply step-by-step counting logic: Work through each position sequentially, calculating available choices while accounting for previously used digits.
4. Calculate final result using multiplication principle: Multiply the number of choices for each position to get the total number of possible identification numbers.

Execution of Strategic Approach

1. Translate the problem requirements

Let's break down what we need to create: identification numbers for employees.

Think of it like creating license plates, but with specific rules:

- Each number must have exactly 4 digits
- All 4 digits must be different (no repeating digits like 1123)
- We can use any digit from 0, 1, 2, 3, 4, 5, 6, 7, 8, 9
- The first digit cannot be 0 (so we won't have numbers like 0123)

Our goal is to count how many different identification numbers we can create following these rules.

Process Skill: TRANSLATE - Converting the word problem into clear mathematical constraints

2. Analyze position-by-position constraints

Let's think about this step by step, imagining we're filling in each position of our 4-digit number: _ _ _ _

For the first position (leftmost digit):
- We cannot use 0
- We can use: 1, 2, 3, 4, 5, 6, 7, 8, 9
- That's 9 possible choices

For the second position:
- We can use any digit from 0-9, BUT
- We cannot repeat the digit we already used in the first position
- Since we used 1 digit already, we have \(10 - 1 = 9\) remaining choices

For the third position:
- We can use any digit from 0-9, BUT
- We cannot repeat either of the 2 digits we've already used
- Since we used 2 digits already, we have \(10 - 2 = 8\) remaining choices

For the fourth position:
- We can use any digit from 0-9, BUT
- We cannot repeat any of the 3 digits we've already used
- Since we used 3 digits already, we have \(10 - 3 = 7\) remaining choices

Process Skill: APPLY CONSTRAINTS - Systematically considering how each restriction affects our choices

3. Apply step-by-step counting logic

Let's work through a concrete example to make this clear:

Suppose we choose 5 for the first position: 5 _ _ _
- First position: 5 (we had 9 choices: 1,2,3,4,5,6,7,8,9)

For the second position, we can use any digit except 5: 5 _ _ _
- Available digits: 0,1,2,3,4,6,7,8,9 (that's 9 choices)
- Let's say we choose 3: 5 3 _ _

For the third position, we can use any digit except 5 and 3: 5 3 _ _
- Available digits: 0,1,2,4,6,7,8,9 (that's 8 choices)
- Let's say we choose 0: 5 3 0 _

For the fourth position, we can use any digit except 5, 3, and 0: 5 3 0 _
- Available digits: 1,2,4,6,7,8,9 (that's 7 choices)
- Let's say we choose 7: 5 3 0 7

This gives us the identification number 5307. The key insight is that no matter which specific digits we choose, the number of available choices for each position follows the same pattern: 9, then 9, then 8, then 7.

4. Calculate final result using multiplication principle

Now we can find the total number of possible identification numbers.

The multiplication principle tells us: if we have multiple independent choices to make, we multiply the number of options for each choice.

In our case:
- First position: 9 choices
- Second position: 9 choices
- Third position: 8 choices
- Fourth position: 7 choices

Total number of identification numbers = \(9 \times 9 \times 8 \times 7\)

Let's calculate step by step:
- \(9 \times 9 = 81\)
- \(81 \times 8 = 648\)
- \(648 \times 7 = 4,536\)

Therefore, there are 4,536 different possible identification numbers.

Final Answer

The answer is 4,536 different identification numbers.

Looking at our answer choices, this matches choice B: 4,536.

We can verify this makes sense: we're selecting 4 different digits with the first digit restricted (no 0), so we expect fewer possibilities than if all positions were unrestricted, but still a substantial number given we have 10 digits to work with.

Common Faltering Points

Errors while devising the approach

1. Misinterpreting the "first digit cannot be 0" constraint

Students often misread this constraint and think that 0 cannot be used anywhere in the 4-digit number. This leads them to believe they can only use digits 1-9 throughout, giving them 9 choices for each position instead of recognizing that 0 can be used in positions 2, 3, and 4.

2. Overlooking the "different digits" requirement

Many students focus on the "first digit cannot be 0" rule but completely miss that all four digits must be different. They might calculate \(9 \times 10 \times 10 \times 10\), allowing repetition of digits, which significantly overcounts the possibilities.

3. Confusing this with a permutation formula approach

Some students recognize this as a permutation problem but incorrectly try to apply \(\mathrm{P}(10,4) = \frac{10!}{(10-4)!} = 5,040\), forgetting about the restriction that the first digit cannot be 0. This leads them to the wrong answer choice C.

Errors while executing the approach

1. Incorrect counting for the second position

After correctly identifying 9 choices for the first position, students might think the second position has 10 choices (forgetting that one digit is already used) or think it has only 8 choices (incorrectly assuming 0 is still unavailable). The correct count is 9 choices for the second position.

2. Arithmetic errors in the final multiplication

Even with the correct setup of \(9 \times 9 \times 8 \times 7\), students often make calculation mistakes. Common errors include: \(9 \times 9 = 72\) (instead of 81), or \(81 \times 8 = 648\) is correct but then \(648 \times 7 = 4,446\) (instead of 4,536).

Errors while selecting the answer

No likely faltering points - the calculation directly leads to 4,536, which clearly matches answer choice B. Students who have executed the approach correctly are unlikely to make selection errors at this stage.