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Let's break down the given problems:

### Part (a):
You are given the expression \( \frac{n!}{(n - r)!} \) and asked to simplify it for \( n = 11 \) and \( r = 5 \), where the result is already provided as 55,440. We need to verify and simplify this expression.

#### Solution:
- \( n! = 11! = 11 \times 10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 39,916,800 \)
- \( (n - r)! = (11 - 5)! = 6! = 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 720 \)

Now divide:
\[
\frac{11!}{(11 - 5)!} = \frac{39,916,800}{720} = 55,440
\]

So, the simplified result for part (a) is **55,440**, which matches the provided result.

---

### Part (b):
You are given the expression \( \frac{n!}{r!(n - r)!} \) and need to simplify it for \( n = 11 \) and \( r = 5 \).

#### Solution:
We already know:
- \( 11! = 39,916,800 \)
- \( (11 - 5)! = 6! = 720 \)
- \( r! = 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \)

Now simplify:
\[
\frac{11!}{r!(n - r)!} = \frac{39,916,800}{120 \times 720} = \frac{39,916,800}{86,400} = 462
\]

So, the simplified result for part (b) is **462**.

---

Do you want further details or have any questions about these calculations?

### Related Questions:
1. How is factorial defined, and what are its properties?
2. What is the difference between a combination and a permutation?
3. How do you calculate combinations using factorials?
4. What is the significance of \( r!(n - r)! \) in the combination formula?
5. Can you solve a similar problem for different values of \( n \) and \( r \)?

**Tip**: When working with factorials, especially large values, be sure to break down calculations step by step to avoid mistakes.

Given the expressions involving factorials, simplify (a) \( \frac{n!}{(n - r)!} \) and (b) \( \frac{n!}{r!(n - r)!} \) where \( n = 11 \) and \( r = 5 \).

Learn how to simplify factorial-based expressions for permutation and combination problems when n=11 and r=5. This includes detailed steps for simplifying \( \frac{n!}{(n - r)!} \) and \( \frac{n!}{r!(n - r)!} \), with solutions 55,440 and 462, respectively.

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