To find the probability of hanging the blue painting first and then the green painting, we can calculate it as follows:
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Probability of hanging the blue painting first: Since there are 5 paintings, the probability of selecting the blue painting first is \( \frac{1}{5} \).
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Probability of hanging the green painting second: After hanging the blue painting, there are 4 paintings left (red, yellow, green, purple). The probability of selecting the green painting next is \( \frac{1}{4} \).
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Combined Probability: The combined probability of both events occurring (blue first and green second) is the product of the individual probabilities:
\[ P(\text{Blue first and Green second}) = P(\text{Blue first}) \times P(\text{Green second}) = \frac{1}{5} \times \frac{1}{4} = \frac{1}{20}. \]
Thus, the correct response is:
\[ \frac{1}{5} \times \frac{1}{4}. \]