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.