To solve the problem, we need to determine how many different combinations of fruits Juan can choose for his fruit salad, given that it must contain watermelon and a total of five different fruits.
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List the fruits: Juan has the following fruits:
- Grapes
- Watermelon
- Apples
- Pineapple
- Bananas
- Mangoes
- Honeydew
- Cantaloupe
This gives us a total of 8 fruits.
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Select the required fruits: Since watermelon must be included in the fruit salad, we will include watermelon in our selection. This means we need to select 4 more fruits from the remaining options.
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Remaining fruits after selecting watermelon: After including watermelon, the available options for the other selections become:
- Grapes
- Apples
- Pineapple
- Bananas
- Mangoes
- Honeydew
- Cantaloupe
This gives us 7 remaining fruits to choose from.
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Choose 4 fruits from the 7 remaining fruits: We need to calculate the number of combinations of 4 fruits that can be selected from these 7 fruits. The number of combinations can be calculated using the binomial coefficient:
\[ \binom{n}{r} = \frac{n!}{r!(n - r)!} \]
where \( n \) is the total number of items (7 fruits) and \( r \) is the number of items to choose (4 fruits).
We calculate:
\[ \binom{7}{4} = \frac{7!}{4!(7 - 4)!} = \frac{7!}{4! \cdot 3!} \]
Simplifying this gives:
\[ = \frac{7 \times 6 \times 5}{3 \times 2 \times 1} = \frac{210}{6} = 35 \]
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Conclusion: The number of ways Juan can make the fruit salad, ensuring that watermelon is one of the fruits and selecting 4 additional fruits from the remaining 7, is \( \boxed{35} \).
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