To solve the problem, we first need to calculate the total number of fruits in the basket and the number of ways to select one apple and one banana.
Step 1: Calculate the total number of fruits.
- Number of bananas = 5
- Number of mangoes = 8
- Number of apples = 7
Total number of fruits = 5 + 8 + 7 = 20 fruits
Step 2: Calculate the number of ways to select one apple and one banana. We need to consider two cases for selecting one apple and one banana: selecting the apple first or selecting the banana first.
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Selecting an apple first then a banana:
- The probability of selecting an apple first = \(\frac{7}{20}\)
- After selecting an apple, there are now 19 fruits left, and the number of bananas remains 5.
- The probability of selecting a banana after an apple = \(\frac{5}{19}\)
So, the probability of selecting an apple first and then a banana = \[ \frac{7}{20} \times \frac{5}{19} \]
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Selecting a banana first then an apple:
- The probability of selecting a banana first = \(\frac{5}{20} = \frac{1}{4}\)
- After selecting a banana, there are now 19 fruits left, and the number of apples remains 7.
- The probability of selecting an apple after a banana = \(\frac{7}{19}\)
So, the probability of selecting a banana first and then an apple = \[ \frac{5}{20} \times \frac{7}{19} = \frac{1}{4} \times \frac{7}{19} \]
Step 3: Combine the probabilities: \[ P(\text{1 apple and 1 banana}) = \left( \frac{7}{20} \times \frac{5}{19} \right) + \left( \frac{5}{20} \times \frac{7}{19} \right) \] \[ = \frac{7 \times 5}{20 \times 19} + \frac{5 \times 7}{20 \times 19} \] \[ = \frac{35}{380} + \frac{35}{380} = \frac{70}{380} = \frac{7}{38} \]
Now let’s check the answer options to find the simplest form corresponding to our calculated fraction \(\frac{7}{38}\).
The possible responses given are:
- \( \frac{7}{80} \)
- \( \frac{21}{190} \)
- \( \frac{7}{76} \)
- \( 35 \)
None of these options correspond to \(\frac{7}{38}\).
Therefore, it looks like the correct calculation yields a different fraction, and if there is a mistake in the formulation of choices, I recommend further clarification on the probability scenario.