To find the probability of selecting 2 carnations from the flower vase containing 5 begonias, 6 carnations, and 7 tulips, we first determine the total number of flowers and then the total ways to select 2 carnations.
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Total number of flowers: \[ 5 \text{ (begonias)} + 6 \text{ (carnations)} + 7 \text{ (tulips)} = 18 \text{ flowers.} \]
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Total ways to choose 2 flowers from 18: The number of ways to choose 2 flowers out of 18 can be calculated using the combination formula \(\binom{n}{k} = \frac{n!}{k!(n-k)!}\): \[ \binom{18}{2} = \frac{18 \times 17}{2 \times 1} = 153. \]
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Total ways to choose 2 carnations from 6: Similarly, the number of ways to choose 2 out of 6 carnations is: \[ \binom{6}{2} = \frac{6 \times 5}{2 \times 1} = 15. \]
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Probability of selecting 2 carnations: The probability \(P\) of selecting 2 carnations is given by the number of ways to choose 2 carnations divided by the total number of ways to choose 2 flowers: \[ P(\text{2 carnations}) = \frac{\text{Number of ways to choose 2 carnations}}{\text{Total ways to choose 2 flowers}} = \frac{15}{153}. \]
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Simplifying the fraction: We can simplify \(\frac{15}{153}\) by finding 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 2 carnations is: \[ \boxed{\frac{5}{51}}. \]