Question

in how many ways can the first 4 guests arrive at a party if 15 guests have been invited?

\\{}_{n}p_{r} = \frac{n!}{(n-r)!}\\

Explanation:

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<reasoning>

Identify the permutation parameters

\[

$$\begin{aligned} n &= 15 \\ r &= 4 \end{aligned}$$

\]

Apply the permutation formula

\[

$$\begin{aligned} {}_{15}P_{4} &= \frac{15!}{(15-4)!} \\ &= \frac{15!}{11!} \end{aligned}$$

\]

Calculate the final value

\[

$$\begin{aligned} {}_{15}P_{4} &= 15 \times 14 \times 13 \times 12 \\ &= 32760 \end{aligned}$$

\]
</reasoning>

<answer>
32760
</answer>

<post_analysis>
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"subject": "Mathematics",
"question_type": "Free Response",
"knowledge_point": [
"Mathematics",
"Algebra",
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Answer:

<pre_analysis>
{
"quality": "clear",
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"skills_matched": [
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],
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</pre_analysis>

<reasoning>

Identify the permutation parameters

\[

$$\begin{aligned} n &= 15 \\ r &= 4 \end{aligned}$$

\]

Apply the permutation formula

\[

$$\begin{aligned} {}_{15}P_{4} &= \frac{15!}{(15-4)!} \\ &= \frac{15!}{11!} \end{aligned}$$

\]

Calculate the final value

\[

$$\begin{aligned} {}_{15}P_{4} &= 15 \times 14 \times 13 \times 12 \\ &= 32760 \end{aligned}$$

\]
</reasoning>

<answer>
32760
</answer>

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"question_type": "Free Response",
"knowledge_point": [
"Mathematics",
"Algebra",
"Permutations"
]
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