To calculate the probability of selecting an apple and a banana from the basket of fruits, we can follow these steps:
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Total number of fruits:
- Bananas: 5
- Mangoes: 8
- Apples: 7
- Total = 5 + 8 + 7 = 20 fruits
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Total combinations of selecting 2 fruits: The total number of ways to choose 2 fruits from 20 is given by the combination formula \( \binom{n}{r} \), where \( n \) is the total number of fruits and \( r \) is the number of fruits to choose.
\[ \text{Total combinations} = \binom{20}{2} = \frac{20 \times 19}{2 \times 1} = 190 \]
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Calculate the combinations for selecting 1 apple and 1 banana:
- Ways to choose 1 apple from 7: \( \binom{7}{1} = 7 \)
- Ways to choose 1 banana from 5: \( \binom{5}{1} = 5 \)
Therefore, the total combinations for selecting 1 apple and 1 banana is:
\[ \text{Combinations for 1 apple and 1 banana} = 7 \times 5 = 35 \]
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Calculate the probability: To find the probability of selecting 1 apple and 1 banana, we divide the combinations of selecting 1 apple and 1 banana by the total combinations of selecting 2 fruits:
\[ \text{Probability} = \frac{\text{Combinations for 1 apple and 1 banana}}{\text{Total combinations}} = \frac{35}{190} \]
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Simplifying the probability: To simplify \( \frac{35}{190} \), we can divide both the numerator and the denominator by 5:
\[ \frac{35 \div 5}{190 \div 5} = \frac{7}{38} \]
Thus, the probability of selecting an apple and a banana is \( \frac{7}{38} \).