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.