Question

a lock has a 4-digit code. each digit can be any number from 0-9. how many different codes are possible if digits can be repeated?

number of codes = ?

Explanation:

<pre_analysis>
{
"quality": "clear",
"question_count": 1,
"skills_matched": [
"step_cot",
"blank_filling_formatter",
"explain_then_answer",
"quick"
],
"needs_drawing": false,
"concepts_used": [
"Fundamental Counting Principle"
],
"new_concepts": [],
"current_concepts": [
"Fundamental Counting Principle",
"Permutations with Repetition"
]
}
</pre_analysis>

<reasoning>

Identify the number of choices per digit

\[
\text{Digits available: } \{0, 1, 2, 3, 4, 5, 6, 7, 8, 9\} \implies 10 \text{ choices}
\]

Apply the Fundamental Counting Principle

\[
\text{Total codes} = 10 \times 10 \times 10 \times 10 = 10^4
\]

Calculate the final value

\[
10^4 = 10000
\]
</reasoning>

<answer>
A lock has a 4-digit code. Each digit can be any number from 0-9. How many different codes are possible if digits can be repeated?

Number of Codes = <blank>10000</blank>
</answer>

<post_analysis>
{
"subject": "Mathematics",
"question_type": "Fill-in-the-blank",
"knowledge_point": [
"Mathematics",
"Algebra",
"Fundamental Counting Principle"
]
}
</post_analysis>

Answer:

<pre_analysis>
{
"quality": "clear",
"question_count": 1,
"skills_matched": [
"step_cot",
"blank_filling_formatter",
"explain_then_answer",
"quick"
],
"needs_drawing": false,
"concepts_used": [
"Fundamental Counting Principle"
],
"new_concepts": [],
"current_concepts": [
"Fundamental Counting Principle",
"Permutations with Repetition"
]
}
</pre_analysis>

<reasoning>

Identify the number of choices per digit

\[
\text{Digits available: } \{0, 1, 2, 3, 4, 5, 6, 7, 8, 9\} \implies 10 \text{ choices}
\]

Apply the Fundamental Counting Principle

\[
\text{Total codes} = 10 \times 10 \times 10 \times 10 = 10^4
\]

Calculate the final value

\[
10^4 = 10000
\]
</reasoning>

<answer>
A lock has a 4-digit code. Each digit can be any number from 0-9. How many different codes are possible if digits can be repeated?

Number of Codes = <blank>10000</blank>
</answer>

<post_analysis>
{
"subject": "Mathematics",
"question_type": "Fill-in-the-blank",
"knowledge_point": [
"Mathematics",
"Algebra",
"Fundamental Counting Principle"
]
}
</post_analysis>