shasadou wrote:
In a business school case competition, the top three teams receive cash prizes of $5,000, $3,000, and $2,000, respectively, while the remaining teams are not ranked and do not receive any prizes. There are 6 participating teams, named Team A, Team B, Team C, Team D, Team E, and Team F. If Team A wins one of the prizes, Team B will also win one of the prizes. How many outcomes of the competition are possible?
A. 18
B. 20
C. 54
D. 84
E. 120
We have two cases to consider: 1) A is one of the top three teams, and 2) A is not one of the top three teams.
Case 1: A is one of the top three teams
If A is one of the top three teams, then B is also one of the top three teams. We only have 4C1 = 4 ways to choose the third top team. In other words, we have 4 possible sets of top three teams (or winning teams). However, for each set of 3 winning teams, there are 3! = 6 ways for how they win the prizes. Therefore, there are 4 x 6 = 24 possible outcomes of the competition if A is one of the top three teams.
Case 2: A is not one of the top three teams
If A is not one of the top three teams, we could have 5C3 = 10 ways to choose the top three teams. (Note that Team B could be one of the top three teams, with Team A NOT being in the top three.) In other words, we have 10 possible sets of top three teams (or winning teams). Similar to case 1, for each set of 3 winning teams, there are 3! = 6 ways for how they win the prizes. Therefore, there are 10 x 6 = 60 possible outcomes of the competition if A is not one of the top three teams.
Therefore, there are a total of 24 + 60 = 84 possible outcomes of the competition.
Answer: D
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