To determine the probability of selecting two carnations from the flower vase, we first need to find the total number of flowers and then calculate the number of favorable outcomes.
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Total number of flowers: \[ \text{Total flowers} = 5 \text{ (begonias)} + 6 \text{ (carnations)} + 7 \text{ (tulips)} = 18 \text{ flowers} \]
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Calculating the number of ways to choose 2 carnations:
- The number of ways to choose 2 carnations from the 6 available is given by the combination formula \( \binom{n}{r} \): \[ \binom{6}{2} = \frac{6!}{2!(6-2)!} = \frac{6 \times 5}{2 \times 1} = 15 \]
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Calculating the total number of ways to choose 2 flowers from 18: \[ \binom{18}{2} = \frac{18!}{2!(18-2)!} = \frac{18 \times 17}{2 \times 1} = 153 \]
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Calculating the probability of selecting 2 carnations: The probability \( P \) is the number of favorable outcomes over the total outcomes: \[ P(\text{2 carnations}) = \frac{\text{Number of ways to choose 2 carnations}}{\text{Total ways to choose 2 flowers}} = \frac{15}{153} \]
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Simplifying the fraction: To simplify \( \frac{15}{153} \), we find the greatest common divisor (GCD) of 15 and 153, which is 3. \[ \frac{15 \div 3}{153 \div 3} = \frac{5}{51} \]
Thus, the probability of selecting two carnations is: \[ \boxed{\frac{5}{51}} \]