The offices of president, vice president, seceretary, and treasurer for a club will be filled from a pool of 12 candidates. Five of them are members of the debate team. What is the probability that all the offices are filled by memebers of the debate team?
5/12*4/11*3/10*2/9
or: 5!(12-5)!/(12!(5-4)!)
567
To find out the probability that all the offices are filled by members of the debate team, we need to calculate the probability of each office being filled by a debate team member, and then multiply these probabilities together.
There are 12 candidates in total, and 5 of them are members of the debate team.
For the first office, the probability of a debate team member being selected is 5/12.
Once the first office has been filled by a debate team member, there are now 11 candidates left, and 4 debate team members remaining.
For the second office, the probability of a debate team member being selected is now 4/11.
Continuing this pattern, for the third office, the probability of a debate team member being selected is 3/10.
Finally, for the fourth office, the probability of a debate team member being selected is 2/9.
To calculate the overall probability, we multiply these individual probabilities together:
(5/12) * (4/11) * (3/10) * (2/9) = 0.0182
Therefore, the probability that all the offices are filled by members of the debate team is 0.0182, or approximately 1.82%.
To find the probability that all the offices are filled by members of the debate team, we need to calculate the number of favorable outcomes and divide it by the total number of possible outcomes.
Step 1: Calculate the number of favorable outcomes.
Since there are 5 members from the debate team and 4 offices to be filled, the first office can be filled by any of the 5 members, the second office can be filled by any of the remaining 4 members, the third office by any of the remaining 3 members, and the fourth office by any of the remaining 2 members. Thus, the number of favorable outcomes is 5 * 4 * 3 * 2 = 120.
Step 2: Calculate the total number of possible outcomes.
Since there are 12 candidates in total, the first office can be filled by any of the 12 candidates, the second office by any of the remaining 11 candidates, the third office by any of the remaining 10 candidates, and the fourth office by any of the remaining 9 candidates. Thus, the total number of possible outcomes is 12 * 11 * 10 * 9 = 11,880.
Step 3: Calculate the probability.
To find the probability, we divide the number of favorable outcomes by the total number of possible outcomes:
Probability = Number of Favorable Outcomes / Total Number of Possible Outcomes
Probability = 120 / 11,880
Therefore, the probability that all the offices are filled by members of the debate team is 120 / 11,880, which can be simplified as 1 / 99 or approximately 0.0101.