To determine the probability of selecting an apple and a banana from the basket containing fruits, we need to calculate the total number of possible selections and the number of favorable outcomes.
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Total number of fruits in the basket:
- Bananas: 5
- Mangoes: 8
- Apples: 7
- Total fruits = 5 + 8 + 7 = 20
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Total ways to select 2 fruits from 20: \[ \text{Total combinations} = \binom{20}{2} = \frac{20 \times 19}{2 \times 1} = 190 \]
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Ways to select 1 apple and 1 banana:
- Ways to choose 1 apple from 7: \(\binom{7}{1} = 7\)
- Ways to choose 1 banana from 5: \(\binom{5}{1} = 5\)
- Total ways to choose 1 apple and 1 banana: \(7 \times 5 = 35\)
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Calculating the probability: \[ P(\text{1 apple and 1 banana}) = \frac{\text{Number of favorable outcomes}}{\text{Total outcomes}} = \frac{35}{190} \]
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Simplifying the fraction:
- Both 35 and 190 can be divided by 5: \[ \frac{35}{190} = \frac{7}{38} \]
None of the provided responses match \(\frac{7}{38}\), so let's confirm our approach and check if there's any mistake in interpreting the selection.
To select one apple first and one banana second (and vice versa) must yield consistent calculations:
- If we calculate:
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Apple first, then Banana:
- \(\text{Probability of 1st being apple} = \frac{7}{20}\)
- \(\text{Probability of 2nd being banana} = \frac{5}{19}\)
- Combined: \(\frac{7}{20} \times \frac{5}{19} = \frac{35}{380}\)
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Banana first, then Apple:
- \(\text{Probability of 1st being banana} = \frac{5}{20}\)
- \(\text{Probability of 2nd being apple} = \frac{7}{19}\)
- Combined: \(\frac{5}{20} \times \frac{7}{19} = \frac{35}{380}\)
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Adding both parts gives us: \[ \frac{35}{380} + \frac{35}{380} = \frac{70}{380} = \frac{7}{38} \]
Thus, the correct simplified probability is \(\frac{7}{38}\). Since it appears that the choices provided do not include this value, it's worth rechecking the selections or any possible mismatches in the answer set provided.
However, based on the calculations: The final calculated probability of selecting one apple and one banana is \(\frac{70}{380}\) or \(\frac{7}{38}\). But from the choices given, none align correctly to the answer we've obtained, which should warrant verification or might suggest a typographical error in options provided.