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In a business school case competition, the top three teams

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by BTGmoderatorDC » Thu Nov 29, 2018 3:35 pm

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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
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by GMATGuruNY » Thu Nov 29, 2018 4:08 pm
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
Case 1: Team A wins a prize, with the result that Team B must win a prize
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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by fskilnik@GMATH » Fri Nov 30, 2018 5:22 pm
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
$$?\,\,\, = \underbrace {C\left( {4,1} \right) \cdot 3!}_{\,\,\,\,\,\,{\text{A}}\,{\text{yes}}{\text{,}}\,\,{\text{B}}\,\,{\text{too}}} + \,\,\,\,\underbrace {C\left( {5,3} \right) \cdot 3!}_{A\,\,{\text{no}}}\,\,\,\, = \,\,\,\,24 + 60\,\,\,\, = \,\,\,\,84$$

This solution follows the notations and rationale taught in the GMATH method.

Regards,
Fabio.
Fabio Skilnik :: GMATH method creator ( Math for the GMAT)
English-speakers :: https://www.gmath.net
Portuguese-speakers :: https://www.gmath.com.br
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by Scott@TargetTestPrep » Thu Apr 04, 2019 5:17 pm
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
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 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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