To find the probability of selecting a $10 bill and a $50 bill when drawing 2 bills without replacement, we start by determining the total number of bills and the desired outcome.
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Count the total number of bills:
- $10 bills: 5
- $20 bills: 10
- $50 bills: 15
- Total bills: 5 + 10 + 15 = 30
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Determine the total outcomes when selecting 2 bills: The number of ways to choose 2 bills from 30 can be calculated using combinations: \[ \binom{30}{2} = \frac{30 \times 29}{2} = 435 \]
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Determine the favorable outcomes for selecting a $10 bill and a $50 bill:
- The number of ways to choose 1 $10 bill from 5: \[ \binom{5}{1} = 5 \]
- The number of ways to choose 1 $50 bill from 15: \[ \binom{15}{1} = 15 \]
- Therefore, the total number of ways to choose 1 $10 bill and 1 $50 bill is: \[ 5 \times 15 = 75 \]
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Calculate the probability: The probability \( P \) of selecting a $10 bill and a $50 bill is given by the ratio of the number of favorable outcomes to the total outcomes: \[ P = \frac{\text{Number of favorable outcomes}}{\text{Total outcomes}} = \frac{75}{435} \]
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Simplify the fraction: To simplify \( \frac{75}{435} \), we find the greatest common divisor (GCD) of 75 and 435, which is 15. \[ \frac{75 \div 15}{435 \div 15} = \frac{5}{29} \]
Thus, the probability of selecting a $10 bill and a $50 bill is: \[ \boxed{\frac{5}{29}} \]