Question

Juan is making a fruit salad. He has grapes, watermelon, apples, pineapple, bananas, mangoes, honeydew, and cantaloupe. He wants his fruit salad to contain five different fruits. How many ways can he make the fruit salad if it must contain watermelon?

21
35
56
70

Answer 1

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.

1. **List the fruits**: Juan has the following fruits:
- Grapes
- Watermelon
- Apples
- Pineapple
- Bananas
- Mangoes
- Honeydew
- Cantaloupe

This gives us a total of 8 fruits.

2. **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.

3. **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.

4. **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:

\[
= \frac{7 \times 6 \times 5}{3 \times 2 \times 1} = \frac{210}{6} = 35
\]

5. **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 \( \boxed{35} \).