To calculate the probability of selecting an apple and a banana from the basket, we'll follow these steps:
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Determine the total number of fruits: \[ \text{Total fruits} = 5 \text{ (bananas)} + 8 \text{ (mangoes)} + 7 \text{ (apples)} = 20 \text{ fruits} \]
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Calculate the number of favorable outcomes for selecting an apple and a banana. There are two scenarios to consider:
- Selecting a banana first and then an apple.
- Selecting an apple first and then a banana.
Let's calculate each:
Scenario 1: Selecting a banana first and then an apple:
- Probability of selecting a banana first: \[ P(\text{Banana first}) = \frac{5}{20} \]
- After selecting a banana, there are now 19 fruits remaining and 7 apples left: \[ P(\text{Apple second | Banana first}) = \frac{7}{19} \]
- Therefore, the probability for this scenario is: \[ P(\text{Banana first and then Apple}) = \frac{5}{20} \times \frac{7}{19} = \frac{5 \cdot 7}{20 \cdot 19} = \frac{35}{380} \]
Scenario 2: Selecting an apple first and then a banana:
- Probability of selecting an apple first: \[ P(\text{Apple first}) = \frac{7}{20} \]
- After selecting an apple, there are 19 fruits remaining and 5 bananas left: \[ P(\text{Banana second | Apple first}) = \frac{5}{19} \]
- Therefore, the probability for this scenario is: \[ P(\text{Apple first and then Banana}) = \frac{7}{20} \times \frac{5}{19} = \frac{7 \cdot 5}{20 \cdot 19} = \frac{35}{380} \]
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Add the probabilities of both scenarios: \[ P(\text{Apple and Banana}) = \frac{35}{380} + \frac{35}{380} = \frac{70}{380} \]
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Simplify the fraction: \[ \frac{70}{380} = \frac{7}{38} \]
Thus, the probability of selecting an apple and a banana is \(\frac{7}{38}\).
Now, reviewing the given responses, none match this probability directly. However, if simplifying further, we recognize that there could have been contextual adjustments or misinterpretations in provided options.
None of the provided responses are equivalent to \(\frac{7}{38}\).
If you are looking for an answer amongst the provided options purely based on probabilities, you would need to ensure the match with any presented probabilities or re-evaluate the problem context.