The notation $P(12, 5)$ represents a permutation. Specifically, it is the number of ways to arrange 5 items from a set of 12 distinct items. The formula for calculating permutations is:

$P(n, r) = \frac{n!}{(n - r)!}$

Where:

• $n$ is the total number of items (12 in this case),
• $r$ is the number of items to arrange (5 in this case),
• $n!$ is the factorial of $n$, and
• $(n - r)!$ is the factorial of $n - r$.

Step-by-step calculation:

$P(12, 5) = \frac{12!}{(12 - 5)!} = \frac{12!}{7!}$

We know that: $12! = 12 \times 11 \times 10 \times 9 \times 8 \times 7!$ Thus, the $7!$ terms cancel out, leaving: $P(12, 5) = 12 \times 11 \times 10 \times 9 \times 8$

Now, multiplying these values: $12 \times 11 = 132$ $132 \times 10 = 1320$ $1320 \times 9 = 11880$ $11880 \times 8 = 95040$

So, $P(12, 5) = 95,040$.

Final Answer:

$P(12, 5) = 95,040$

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Here are some related questions you might find interesting:

1. How is the formula for permutations derived?
2. What is the difference between permutations and combinations?
3. How do you calculate combinations using the formula $C(n, r)$?
4. How can we apply permutations in real-life problems, like arranging a set of books on a shelf?
5. What are some examples of when order matters in probability problems?

Tip: Always remember that in permutations, the order of selection does matter!