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
OA D
Source: Veritas Prep
In a business school case competition, the top three teams
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Case 1: Team A wins a prize, with the result that Team B must win a prizeBTGmoderatorDC 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
Number of prizes that could be won by A = 3.
Number of remaining prizes that could be won by B = 2.
Number of teams that could win the third prize = 4. (Any of the 4 remaining teams.)
To combine these options, we multiply:
3*2*4 = 24.
Case 2: Team A does NOT win a prize
Number of teams that could win the first prize = 5. (Of the 6 teams, any team but A.)
Number of teams that could win the second prize = 4. (Of the 5 remaining teams, any team but A.)
Number of teams that could win the third prize = 3. (Of the 4 remaining teams, any team but A.)
To combine these options, we multiply:
5*4*3 = 60.
Total ways = Case 1 + Case 2 = 24 + 60 = 84.
The correct answer is D.
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$$?\,\,\, = \underbrace {C\left( {4,1} \right) \cdot 3!}_{\,\,\,\,\,\,{\text{A}}\,{\text{yes}}{\text{,}}\,\,{\text{B}}\,\,{\text{too}}} + \,\,\,\,\underbrace {C\left( {5,3} \right) \cdot 3!}_{A\,\,{\text{no}}}\,\,\,\, = \,\,\,\,24 + 60\,\,\,\, = \,\,\,\,84$$BTGmoderatorDC 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
Source: Veritas Prep
This solution follows the notations and rationale taught in the GMATH method.
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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.BTGmoderatorDC 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
OA D
Source: Veritas Prep
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 have only 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 the order in which 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, even if Team A is not 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 the order in which 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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