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20 Dec 2024, 09:00
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12 Days of Christmas 2024 - 2025 Competition with $40,000 of Prizes
At a Football Championship, 5 players from the Spanish team each scored at least one goal. Did any of them score at least 5 goals?
(1) Together, the 5 players scored a total of 20 goals during the tournament.
(2) No two of the 5 players scored the same number of goals.
At a Football Championship, 5 players from the Spanish team each scored at least one goal. Did any of them score at least 5 goals?
(1) Together, the 5 players scored a total of 20 goals during the tournament.
(2) No two of the 5 players scored the same number of goals.
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20 Dec 2024, 10:00
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GMAT Club's Official Explanation:
At a Football Championship, 5 players from the Spanish team each scored at least one goal. Did any of them score at least 5 goals?
(1) Together, the 5 players scored a total of 20 goals during the tournament.
If each of the five players scored 4 goals, then none of them scored 5 or more goals. However, if, for example, 4 players scored 2 goals each and the fifth player scored 12, then 1 player scored at least 5 goals. Not sufficient.
(2) No two of the 5 players scored the same number of goals.
Is it possible for all players to score fewer than 5 goals while still having no two players score the same number of goals? No. If the first four players scored 1, 2, 3, and 4 goals each (all less than 5), the fifth player, having to score a unique number of goals, must score at least 5. Thus, at least one player scored 5 or more goals. Sufficient.
Answer: B.
General Discussion
20 Dec 2024, 09:24
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We need to determine whether any of the 5 players on the Spanish team scored at least 5 goals. Let's evaluate the two statements independently and then together.
Statement (1): Together, the 5 players scored a total of 20 goals during the tournament.
If the total number of goals scored by the 5 players is 20, one possible distribution is:
5 players score 4 goals each:
4+4+4+4+4=20. In this case, no player scored 5 goals.
However, other distributions are also possible:
One player scores 5 goals, and the others score:
5+5+4+3+3=20. In this case, at least one player scored 5 goals.
Since multiple distributions of goals are possible, Statement (1) alone is insufficient.
Statement (2): No two of the 5 players scored the same number of goals.
If no two players scored the same number of goals, the 5 players must have scored distinct numbers of goals. The smallest possible distinct integers for 5 players are
0,1,2,3, and 4.
The total in this case is 0+1+2+3+4=10, which is less than 20.
Thus, for the total to be 20, the 5 distinct numbers must be larger. The next smallest set of 5 distinct integers is 2,3,4,5, and 6. The total in this case is:
2+3+4+5+6=20.
Here, one player (at least) scored 5 goals.
Since the distinctness of scores forces at least one player to score 5 goals if the total is 20, Statement (2) alone is sufficient.
Combining Statements (1) and (2):
While Statement (1) alone is insufficient, combining it with Statement (2) reinforces that the scores must sum to 20 and be distinct, as shown in the analysis of Statement (2). This guarantees that at least one player scored 5 goals. The combination does not add new information beyond Statement (2) alone.
Final Answer: (B)
Statement (2) alone is sufficient, but Statement (1) alone is not.
Statement (1): Together, the 5 players scored a total of 20 goals during the tournament.
If the total number of goals scored by the 5 players is 20, one possible distribution is:
5 players score 4 goals each:
4+4+4+4+4=20. In this case, no player scored 5 goals.
However, other distributions are also possible:
One player scores 5 goals, and the others score:
5+5+4+3+3=20. In this case, at least one player scored 5 goals.
Since multiple distributions of goals are possible, Statement (1) alone is insufficient.
Statement (2): No two of the 5 players scored the same number of goals.
If no two players scored the same number of goals, the 5 players must have scored distinct numbers of goals. The smallest possible distinct integers for 5 players are
0,1,2,3, and 4.
The total in this case is 0+1+2+3+4=10, which is less than 20.
Thus, for the total to be 20, the 5 distinct numbers must be larger. The next smallest set of 5 distinct integers is 2,3,4,5, and 6. The total in this case is:
2+3+4+5+6=20.
Here, one player (at least) scored 5 goals.
Since the distinctness of scores forces at least one player to score 5 goals if the total is 20, Statement (2) alone is sufficient.
Combining Statements (1) and (2):
While Statement (1) alone is insufficient, combining it with Statement (2) reinforces that the scores must sum to 20 and be distinct, as shown in the analysis of Statement (2). This guarantees that at least one player scored 5 goals. The combination does not add new information beyond Statement (2) alone.
Final Answer: (B)
Statement (2) alone is sufficient, but Statement (1) alone is not.
