Rework problem 15 from section 2.3 of your text, involving the vegetable and fruit salads. Assume that you have 8 different types of vegetables and 10 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 2 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 salads?

The number of ways to prepare a vegetable salad and a fruit salad can be found by multiplying the number of ways to choose the vegetables by the number of ways to choose the fruits. The number of ways to prepare either a vegetable salad or a fruit salad but not both can be found by subtracting the number of ways to prepare both salads from the total number of ways to prepare either salad. The number of ways to choose two different salads can be found by adding the number of ways to prepare a vegetable salad to the number of ways to prepare a fruit salad.

Explanation:

In order to find the number of ways to prepare a vegetable salad and a fruit salad, we need to multiply the number of ways to choose the vegetables by the number of ways to choose the fruits. So, there are 8 choose 3 ways to choose the vegetables and 10 choose 2 ways to choose the fruits. Multiplying these two numbers together gives us the total number of ways to prepare both salads.

To find the number of ways to prepare either a vegetable salad or a fruit salad but not both, we need to subtract the number of ways to prepare both salads from the total number of ways to prepare either salad. So, we subtract the product of 8 choose 3 and 10 choose 2 from the sum of 8 choose 3 and 10 choose 2.

To find the number of ways to choose two different salads, we can add the number of ways to prepare a vegetable salad to the number of ways to prepare a fruit salad. So, we add 8 choose 3 to 10 choose 2.

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