To calculate the probability of selecting one apple and one banana from the basket of fruits, we start by determining the total number of fruits and the possible outcomes for selecting 2 fruits.
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Total number of fruits:
- Bananas: 5
- Mangoes: 8
- Apples: 7
- Total = 5 + 8 + 7 = 20 fruits
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Calculating the number of ways to select 1 apple and 1 banana:
- Number of ways to choose 1 apple from 7: \( \binom{7}{1} = 7 \)
- Number of ways to choose 1 banana from 5: \( \binom{5}{1} = 5 \)
- Thus, the total ways to choose 1 apple and 1 banana = \( 7 \times 5 = 35 \)
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Calculating the total number of ways to select 2 fruits from 20:
- Number of ways to choose 2 fruits from 20: \( \binom{20}{2} = \frac{20 \times 19}{2} = 190 \)
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Calculating the probability: \[ P(\text{1 apple and 1 banana}) = \frac{\text{Number of ways to choose 1 apple and 1 banana}}{\text{Total ways to choose 2 fruits}} = \frac{35}{190} \]
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Simplifying the fraction:
- \( \frac{35}{190} = \frac{7}{38} \)
After evaluating the provided answers (35, \(\frac{3}{5}\), \(\frac{7}{76}\), \(\frac{7}{80}\), \( \frac{21}{190}\)), we see that \(\frac{21}{190}\) is equivalent to \( \frac{35}{190} \).
Therefore, the probability of selecting one apple and one banana is:
Answer: \( \frac{21}{190} \).