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Answer :
Final answer:
Using combinations, there are 1225 ways to prepare a vegetable salad and a fruit salad, 70 ways to prepare either a vegetable salad or a fruit salad but not both, and 4830 ways to choose two different salads.
Explanation:
The subject of this question is combinatorics in mathematics. You need to use the formula for combinations to solve each part of the problem, which is nCr = n! / [r!(n-r)!], where 'n' is the total number of items, 'r' is the number of items to choose, and '!' denotes a factorial.
(1) For the vegetable salad, we have 7 vegetables and we need to choose any 3. Using the combinations formula, that gives us 7C3 = 35 ways. Similarly, for the fruit salad, we have 7 fruits and we need to choose any 3, which also gives us 7C3 = 35 ways. Since we are preparing both salads, we multiply these together, which gives us 35*35 = 1225 different ways to prepare a vegetable salad and a fruit salad.
(2) If you need to prepare a vegetable salad or a fruit salad, but not both, you simply add the number of ways to prepare each salad. So, 35 ways for the vegetable salad and 35 ways for the fruit salad gives us 35 + 35 = 70 different ways.
(3) If you are choosing two different salads, and this could be two vegetable salads or two fruit salads or one of each, then we have 70 choices for the first salad and 69 remaining choices for the second salad, so the total number of ways to choose two different salads is 70*69 = 4830 ways.
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