To find the probability of selecting two carnations from the vase, we first determine the total number of flowers in the vase and then the number of ways to select two carnations.
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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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Number of ways to select 2 carnations: The number of ways to choose 2 carnations from 6 can be calculated using the combination formula: \[ \binom{n}{k} = \frac{n!}{k!(n-k)!} \] For our case (choosing 2 from 6): \[ \binom{6}{2} = \frac{6!}{2!(6-2)!} = \frac{6 \times 5}{2 \times 1} = 15 \]
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Total number of ways to select any 2 flowers from 18: Using the combination formula again: \[ \binom{18}{2} = \frac{18!}{2!(18-2)!} = \frac{18 \times 17}{2 \times 1} = 153 \]
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Probability of selecting 2 carnations: The probability is given by the ratio of the number of ways to choose 2 carnations to the total number of ways to choose any 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: To simplify \(\frac{15}{153}\), we find the greatest common divisor: The GCD of 15 and 153 is 3. \[ \frac{15 \div 3}{153 \div 3} = \frac{5}{51} \]
Thus, the probability of selecting two carnations is: \[ \boxed{\frac{5}{51}} \]