Let's break down each part of the problem:

1. Evaluate $\frac{8!}{3!5!}$:

   The expression $\frac{8!}{3!5!}$ is a combination formula, often denoted as $\binom{8}{3}$ or $\binom{8}{5}$, because $\binom{8}{3} = \binom{8}{5}$. It represents the number of ways to choose 3 items out of 8 without regard to order.

   $\frac{8!}{3!5!} = \frac{8 \times 7 \times 6 \times 5!}{3! \times 5!} = \frac{8 \times 7 \times 6}{3 \times 2 \times 1} = 56$

2. Evaluate $P(8, 3)$:

   The permutation formula $P(8, 3)$ represents the number of ways to arrange 3 items out of 8.

   $P(8, 3) = \frac{8!}{(8-3)!} = \frac{8!}{5!} = 8 \times 7 \times 6 = 336$

3. Evaluate $C(8, 5)$:

   The combination formula $C(8, 5)$ is the same as $\binom{8}{5}$.

   $C(8, 5) = \frac{8!}{5!3!} = 56$

4. Evaluate $C(8, 3)$:

   The combination formula $C(8, 3)$ is the same as $\binom{8}{3}$.

   $C(8, 3) = \frac{8!}{3!5!} = 56$

5. Evaluate $P(8, 5)$:

   The permutation formula $P(8, 5)$ represents the number of ways to arrange 5 items out of 8.

   $P(8, 5) = \frac{8!}{(8-5)!} = \frac{8!}{3!} = 8 \times 7 \times 6 \times 5 \times 4 = 6720$

Summarizing the results:

1. $\frac{8!}{3!5!} = 56$
2. $P(8, 3) = 336$
3. $C(8, 5) = 56$
4. $C(8, 3) = 56$
5. $P(8, 5) = 6720$

Do you want details or have any questions?

Related Questions:

1. What is the general formula for permutations $P(n, k)$?
2. How do combinations differ from permutations in terms of order?
3. Can you explain why $C(8, 3)$ is equal to $C(8, 5)$?
4. How would you calculate $C(10, 4)$?
5. What are some practical applications of permutations and combinations?
6. What is the binomial theorem, and how does it relate to combinations?
7. How can permutations be used to solve problems involving seating arrangements?
8. How does factorial notation simplify the calculation of permutations and combinations?

Tip:

When solving problems involving permutations and combinations, it's helpful to write down the formulas and identify what each variable represents. This practice can clarify the steps needed to find the solution.