1. The Rutgers Combinatorics Club consists of 20 students. Out of those 20, 5 of
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Question
1. The Rutgers Combinatorics Club consists of 20 students. Out of those 20, 5 of them are seniors. An executive committee with 5 members for the club needs to be formed, and the club rules require that at least two seniors are on the committee.
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(a) Here is an attempt at a decision process to count the number of committees:
i. Choose two seniors.
ii. Choose 3 more people out of the remaining 18.
Show that this is not a correct decision process for counting the number of possible committees. Specifically, give two sequences of decisions that lead to the same committee.
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(b) Now count the number of possible committees. Explain your answer (with a decision process, or in words, or both).
Explanation / Answer
Part a
In the given scenario, we first choose two seniors and then three more people out of remaining 18. This is not a correct decision process for counting the number of possible committees because we select remaining three people from the remaining 18 people but it would required to select the remaining three from the 15 members who are not senior. Two senior members are already selected from the five senior members. Let us consider the first sequence in which first two seniors are selected and then remaining three members are selected from remaining 18 but there would be probability of selection of remaining three seniors in the committee. Also, same can be happen for the next attempt which leads to the same committee.
Part b
Now, we have to count the number of possible committees.
We can select 2 seniors out of 5 seniors in 5C2 ways.
We can select 3 other members from remaining 15 members in a 15C3 ways.
Required number of ways = 5C2*15C3 = 10*455 = 4550
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