Rework problem 15 from section 2.3 of your text, involving the vegetable and fruit salads. Assume that you have 5 different types of vegetables and 9 different types of fruit. A vegetable salad consists of a mixture of any 3 types of vegetables, and a fruit salad consists of a mixture of 4 kinds of fruit.
(1) In how many different ways can you prepare a vegetable salad and a fruit salad?
(2) In how many different ways can you prepare a vegetable salad or a fruit salad but not both?
(3) How many ways can you choose two different kinds of salad?

Let's go through each part of the problem step by step.

### Given:
- 5 different types of vegetables.
- 9 different types of fruit.
- A vegetable salad consists of a mixture of any 3 types of vegetables.
- A fruit salad consists of a mixture of any 4 types of fruit.

### Part (1): **In how many different ways can you prepare a vegetable salad and a fruit salad?**

For this part, we are looking for the number of ways to prepare a vegetable salad **and** a fruit salad together.

1. **Vegetable Salad**: You need to select 3 vegetables out of 5. The number of ways to do this is given by the binomial coefficient \( \binom{5}{3} \), which represents the number of ways to choose 3 items from a set of 5.
   \[
   \binom{5}{3} = \frac{5!}{3!(5-3)!} = \frac{5 \times 4}{2 \times 1} = 10
   \]

2. **Fruit Salad**: You need to select 4 fruits out of 9. The number of ways to do this is given by \( \binom{9}{4} \).
   \[
   \binom{9}{4} = \frac{9!}{4!(9-4)!} = \frac{9 \times 8 \times 7 \times 6}{4 \times 3 \times 2 \times 1} = 126
   \]

3. **Total combinations for both a vegetable salad and a fruit salad**: Since the selection of vegetable and fruit salads are independent events, the total number of ways to prepare both a vegetable salad and a fruit salad is:
   \[
   \binom{5}{3} \times \binom{9}{4} = 10 \times 126 = 1260
   \]

### Part (2): **In how many different ways can you prepare a vegetable salad or a fruit salad but not both?**

Here, we want the number of ways to prepare either a vegetable salad or a fruit salad, but not both.

1. **Number of ways to prepare a vegetable salad**: From the previous part, the number of ways to prepare just a vegetable salad is \( \binom{5}{3} = 10 \).

2. **Number of ways to prepare a fruit salad**: Similarly, the number of ways to prepare just a fruit salad is \( \binom{9}{4} = 126 \).

3. **Number of ways to prepare either a vegetable salad or a fruit salad but not both**: Since we are looking for "either or" but not both, we simply add the two cases together:
   \[
   10 + 126 = 136
   \]

### Part (3): **How many ways can you choose two different kinds of salad?**

For this part, we are asked to determine how many ways we can choose two different kinds of salad (meaning both a vegetable salad and a fruit salad). This has already been calculated in Part (1), where we found the number of ways to prepare both a vegetable salad and a fruit salad:
   \[
   \text{The number of ways to choose two different kinds of salad} = 1260
   \]

### Final Answers:

1. The number of ways to prepare both a vegetable salad and a fruit salad: **1260**.
2. The number of ways to prepare a vegetable salad or a fruit salad but not both: **136**.
3. The number of ways to choose two different kinds of salad: **1260**.

Would you like more details or have any questions? Here are some related questions to extend your understanding:

1. What if you had to choose 4 vegetables and 3 fruits? How would the solution change?
2. How many ways can you choose 2 types of vegetables from 5?
3. What is the probability of choosing exactly one vegetable salad if you can choose up to 2 types of salads?
4. How does the calculation change if we wanted a mixture of 5 vegetables and 6 fruits?
5. What if there were restrictions on which vegetables or fruits could be mixed together?

**Tip**: Whenever you encounter problems with combinations, it's useful to break down the problem into independent events and multiply the combinations of each.

Explore how to solve a combinatorics problem involving the selection of 3 vegetables from 5 and 4 fruits from 9. This problem includes calculations using the combination formula C(n, k) and covers multiple scenarios such as selecting either one or both types of salads. Suitable for students in Grades 10-12.

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