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# Probability

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There are 3 math clubs in the school district, with 5, 7, and 8 students respectively. Each club has two co-presidents. If I randomly select a club, and then randomly select four members of that club to give a copy of Introduction to Counting and Probability, what is the probability that two of the people who receive books are co-presidents?

Aug 6, 2022

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With the 5 student club, there are $${5 \choose 4} = 5$$ ways to choose the 4 students. Of these, there are $${2 \choose 2} \times {3 \choose 2} = 3$$ ways to select both co-presidents. Both co-presidents and be chosen in 1 way, and there $${3 \choose 2} = 3$$ ways to choose the remaining students. The probability of selecting this club is $${1 \over 3}$$. So, the probability for this club is $${3 \over 5} \times {1 \over 3} = {1 \over 5}$$

With the 7 student club, there are $${7 \choose 4} = 35$$ ways to choose the 4 students. Of these, there are $${2 \choose 2} \times {5 \choose 2} = 5$$ ways to select both co-presidents. Likewise, the probability of selecting this club is $${1 \over 3}$$. So, the probability for this club is $${1 \over 7} \times {1 \over 3} = {1 \over 21}$$

With the club with 8 students, there are $${8 \choose 4} = 70$$ ways to choose the 4 students. There are also $${2 \choose 2} \times {6 \choose 2} = 15$$ ways to choose both co-presidents.

So, the probability is $${3 \over 14} \times {1 \over 3} = {1 \over 14}$$

Adding everything up, we find the total probability is $${1 \over 5} + {1 \over 21} + {1 \over 14} = \color{brown}\boxed{67 \over 210}$$

Aug 6, 2022
edited by BuilderBoi  Aug 6, 2022