The problem in the image asks to evaluate the permutation $12P_5$.

The formula for permutations is:

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

Where:

• $n!$ means "n factorial," the product of all positive integers from 1 to $n$.
• $r!$ means "r factorial."
• $nP_r$ is the number of ways to arrange $r$ objects from a set of $n$.

For this case, $n = 12$ and $r = 5$, so we need to calculate $12P_5$.

$12P_5 = \frac{12!}{(12 - 5)!} = \frac{12!}{7!}$

Now, calculate the value:

$12P_5 = \frac{12 \times 11 \times 10 \times 9 \times 8 \times 7!}{7!}$

The $7!$ cancels out, so we are left with:

$12P_5 = 12 \times 11 \times 10 \times 9 \times 8$

Let's compute that.The value of $12P_5$ is 95,040.

Thus, the correct answer is the first option: 95,040.

Would you like more details on this calculation or have any further questions?

Here are 5 related questions for practice:

1. Evaluate $10P_3$.
2. What is the difference between permutations and combinations?
3. How do you calculate $nP_r$ when $n$ and $r$ are very large?
4. Calculate $7P_4$.
5. If a problem gives $n = 15$ and $r = 6$, what is $15P_6$?

Tip: In permutations, the order of selection matters, unlike combinations where the order does not matter.