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There are 77 students performing in the talent show, 55 of whom will juggle.\newlineIf 44 students are randomly chosen to perform during the first section of the show, what is the probability that all of them will juggle?\newlineWrite your answer as a decimal rounded to four decimal places.\newline____\newline

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Q. There are 77 students performing in the talent show, 55 of whom will juggle.\newlineIf 44 students are randomly chosen to perform during the first section of the show, what is the probability that all of them will juggle?\newlineWrite your answer as a decimal rounded to four decimal places.\newline____\newline
  1. Calculate Probability First Pick: First, calculate the probability of choosing a juggling student on the first pick. There are 55 jugglers out of 77 students.\newlineProbability for the first pick =Number of jugglersTotal number of students=57.= \frac{\text{Number of jugglers}}{\text{Total number of students}} = \frac{5}{7}.
  2. Calculate Probability Second Pick: Next, calculate the probability of choosing a juggling student on the second pick. After one juggler is picked, there are 44 jugglers left and 66 students in total.\newlineProbability for the second pick =Remaining jugglersRemaining students=46.= \frac{\text{Remaining jugglers}}{\text{Remaining students}} = \frac{4}{6}.
  3. Calculate Probability Third Pick: Now, calculate the probability of choosing a juggling student on the third pick. There are now 33 jugglers left and 55 students in total.\newlineProbability for the third pick =Remaining jugglersRemaining students=35.= \frac{\text{Remaining jugglers}}{\text{Remaining students}} = \frac{3}{5}.
  4. Calculate Probability Fourth Pick: Finally, calculate the probability of choosing a juggling student on the fourth pick. There are 22 jugglers left and 44 students in total.\newlineProbability for the fourth pick =Remaining jugglersRemaining students=24.= \frac{\text{Remaining jugglers}}{\text{Remaining students}} = \frac{2}{4}.
  5. Multiply Probabilities: To find the overall probability that all 44 chosen students will juggle, multiply the probabilities of each pick together.\newlineOverall probability = (57)×(46)×(35)×(24)(\frac{5}{7}) \times (\frac{4}{6}) \times (\frac{3}{5}) \times (\frac{2}{4}).
  6. Simplify Multiplication: Simplify the multiplication of fractions by canceling out common factors.\newlineOverall probability = (57)×(23)×(35)×(12)(\frac{5}{7}) \times (\frac{2}{3}) \times (\frac{3}{5}) \times (\frac{1}{2}).
  7. Cancel Common Factors: After canceling out common factors, the remaining multiplication is:\newlineOverall probability = (17)×(13)×(11)×(12)(\frac{1}{7}) \times (\frac{1}{3}) \times (\frac{1}{1}) \times (\frac{1}{2}).
  8. Multiply Remaining Fractions: Now, multiply the remaining fractions to get the final probability.\newlineOverall probability = (17)×(13)×(12)=142(\frac{1}{7}) \times (\frac{1}{3}) \times (\frac{1}{2}) = \frac{1}{42}.

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