To simplify the expression \( A_{15}^4 \), let's start by recalling the definition of the permutation formula:

\[
A_n^r = \frac{n!}{(n-r)!}
\]

For \( A_{15}^4 \), we have \( n = 15 \) and \( r = 4 \). Plugging these values into the formula gives us:

\[
A_{15}^4 = \frac{15!}{(15-4)!} = \frac{15!}{11!}
\]

Now let's simplify the factorials:

\[
15! = 15 \times 14 \times 13 \times 12 \times 11!
\]

So,

\[
\frac{15!}{11!} = \frac{15 \times 14 \times 13 \times 12 \times 11!}{11!} = 15 \times 14 \times 13 \times 12
\]

Thus,

\[
A_{15}^4 = 15 \times 14 \times 13 \times 12
\]

To find the numerical value:

\[
15 \times 14 = 210
\]
\[
210 \times 13 = 2730
\]
\[
2730 \times 12 = 32760
\]

Therefore,

\[
A_{15}^4 = 32760
\]

Would you like more details or have any questions about this solution? Here are five more questions you might be interested in:

1. How do you calculate combinations and permutations?
2. What is the difference between permutations and combinations?
3. How do you simplify factorial expressions?
4. What are some real-life applications of permutations?
5. How do you solve problems involving permutations and combinations?

**Tip:** Understanding the factorial function and its properties is crucial for simplifying and solving problems involving permutations and combinations.

Learn how to simplify the permutation formula A_{15}^4 = \frac{15!}{(15-4)!} step-by-step. This problem involves factorials and permutations, providing a comprehensive understanding of their application. Suitable for students in Grades 9-12.

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