Question

exercise 33: use fundamental counting principle

1. out of a group of eight students serving on the student government association, how many different ways can a president, a vice president, and a treasurer be selected?

Explanation:

<pre_analysis>
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"quality": "clear",
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],
"needs_drawing": false,
"concepts_used": [
"Fundamental Counting Principle",
"Permutations"
],
"new_concepts": [],
"current_concepts": [
"Fundamental Counting Principle",
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}
</pre_analysis>

<reasoning>

Identify the selection stages and available choices

\[

$$\begin{aligned} &\text{Total students } (n) = 8\\ &\text{Positions to fill } (k) = 3 \quad (\text{President, Vice President, Treasurer}) \end{aligned}$$

\]

Apply the Fundamental Counting Principle

\[

$$\begin{aligned} &\text{Ways to choose President} = 8\\ &\text{Ways to choose Vice President} = 7\\ &\text{Ways to choose Treasurer} = 6 \end{aligned}$$

\]

Calculate the total number of ways

\[
\text{Total ways} = 8 \times 7 \times 6 = 336
\]
</reasoning>

<answer>
336
</answer>

<post_analysis>
{
"subject": "Mathematics",
"question_type": "Free Response",
"knowledge_point": [
"Mathematics",
"Algebra",
"Fundamental Counting Principle"
]
}
</post_analysis>

Answer:

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

<reasoning>

Identify the selection stages and available choices

\[

$$\begin{aligned} &\text{Total students } (n) = 8\\ &\text{Positions to fill } (k) = 3 \quad (\text{President, Vice President, Treasurer}) \end{aligned}$$

\]

Apply the Fundamental Counting Principle

\[

$$\begin{aligned} &\text{Ways to choose President} = 8\\ &\text{Ways to choose Vice President} = 7\\ &\text{Ways to choose Treasurer} = 6 \end{aligned}$$

\]

Calculate the total number of ways

\[
\text{Total ways} = 8 \times 7 \times 6 = 336
\]
</reasoning>

<answer>
336
</answer>

<post_analysis>
{
"subject": "Mathematics",
"question_type": "Free Response",
"knowledge_point": [
"Mathematics",
"Algebra",
"Fundamental Counting Principle"
]
}
</post_analysis>