To find the probability of selecting two carnations from the flower vase, we first need to determine the total number of flowers and the total number of ways to choose two flowers.
Step 1: Count the total number of flowers.
- Number of begonias: 5
- Number of carnations: 6
- Number of tulips: 7
Total number of flowers = 5 + 6 + 7 = 18 flowers
Step 2: Calculate the total number of ways to select 2 flowers from the 18. We use the combination formula: \[ \text{Number of ways to choose 2 flowers from 18} = \binom{18}{2} = \frac{18 \times 17}{2 \times 1} = 153 \]
Step 3: Calculate the number of ways to select 2 carnations from the 6 carnations. \[ \text{Number of ways to choose 2 carnations from 6} = \binom{6}{2} = \frac{6 \times 5}{2 \times 1} = 15 \]
Step 4: Calculate the probability of selecting 2 carnations. The probability is given by the ratio of the favorable outcomes (selecting 2 carnations) to the total outcomes (selecting any 2 flowers). \[ P(\text{2 carnations}) = \frac{\text{Number of ways to choose 2 carnations}}{\text{Total number of ways to choose 2 flowers}} = \frac{15}{153} \]
Step 5: Simplify the fraction. Both the numerator and denominator can be divided by 51: \[ \frac{15 \div 51}{153 \div 51} = \frac{5}{51} \]
Thus, the probability of selecting two carnations is: \[ \boxed{\frac{5}{51}} \]