Test: Combinations - GMAT MCQ

Statement (1): Dora sat next to Ethyl and Carl.

This statement provides information about the seating arrangement, but it does not mention anything about Grace or Bill. Therefore, it does not provide any direct information about whether Grace sat next to Bill.

Statement (2): Grace sat next to Carl.

This statement provides information specifically about Grace's seating arrangement, stating that she sat next to Carl. However, it does not provide any information about Bill or their relative positions.

When we consider each statement alone:

• Statement (1) alone tells us about the seating arrangement involving Dora, Ethyl, and Carl, but it does not provide any direct information about Grace or Bill. It does not help us determine if Grace sat next to Bill.
• Statement (2) alone tells us that Grace sat next to Carl, but it does not provide any information about Bill. It does not help us determine if Grace sat next to Bill either.

Therefore, statement (1) alone is sufficient to answer the question because it provides information about the seating arrangement involving Dora, Ethyl, and Carl. Since Dora sat next to Ethyl and Carl, there is no room for Grace to sit next to Bill. Thus, Grace did not sit next to Bill.

The answer is A.

Statement (1): The display contains an equal number of gold and silver coins.

This statement tells us that the number of gold coins is equal to the number of silver coins. However, it does not provide any information about the total number of coins or the specific arrangement of the coins. Therefore, statement (1) alone is not sufficient to answer the question.

Statement (2): If only the silver coins were displayed, 5,040 different arrangements of the silver coins would be possible.

This statement provides information about the number of different arrangements possible for the silver coins only. It does not provide any information about the gold coins or the overall arrangement of all the coins. Therefore, statement (2) alone is not sufficient to answer the question.

When we consider both statements together:

Neither statement provides information about the total number of coins or the specific arrangement of the coins. While statement (1) tells us that the number of gold and silver coins is equal, it doesn't provide any further details.

Therefore, when considered together, statements (1) and (2) are sufficient to answer the question. The answer is C: BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient.

Statement (1): If a team of 3 members is selected from the group of x members, then 30 different combinations of 3 members can be formed with 2 specific members never being together on the team.

This statement tells us that there are 30 different combinations of 3 members that can be formed, and specifically, two specific members are never together in any of these combinations. However, it does not provide the total number of members in the group. Therefore, statement (1) alone is not sufficient to answer the question.

Statement (2): There are a total of 35 ways to make a team of 3 members out of a team of x members.

This statement tells us that there are 35 different ways to form a team of 3 members from the group of x members. However, it does not provide the value of x, the total number of members in the group. Therefore, statement (2) alone is not sufficient to answer the question.

When we consider each statement alone:

• Statement (1) alone does not provide the value of x.
• Statement (2) alone does not provide the value of x.

When we consider both statements together:

• Statement (1) tells us that there are 30 different combinations of 3 members, and statement (2) tells us that there are 35 different ways to form a team of 3 members. Since these two statements provide information about the same scenario, we can combine them to determine the value of x.
• From statement (2), we know that the total number of ways to form a team of 3 members is 35. From statement (1), we know that there are 30 different combinations with two specific members never being together. This means that the remaining 5 combinations must involve these two specific members being together.

Therefore, the value of x is 5 (the two specific members) + 1 (the third member) = 6.

Hence, when considered together, each statement alone is sufficient to answer the question. The answer is D: EACH statement ALONE is sufficient to answer the question asked.

Statement (1): 17 students are aged between 21 and 23.

This statement tells us the number of students in the age range of 21 to 23 but does not provide any information about their scores. Without information about their scores, we cannot determine how many students in this age range scored between 70 and 90.

Statement (2): 17 students scored between 70 and 90.

This statement tells us the number of students who scored between 70 and 90, but it does not provide any information about their ages. Without information about their ages, we cannot determine how many of these students fall into the age range of 21 to 23.

When we consider each statement alone:

• Statement (1) alone is not sufficient to answer the question because it does not provide information about the scores of the students.
• Statement (2) alone is not sufficient to answer the question because it does not provide information about the ages of the students.

Therefore, statements (1) and (2) together are not sufficient to answer the question either. We need additional data, such as the overlap between the students aged between 21 and 23 and those who scored between 70 and 90, to determine the number of students who satisfy both criteria. The answer is E.

Statement (1): All subcommittees must have at least two members.

This statement indicates that each subcommittee must consist of at least two members. With this restriction, we can count the number of subcommittees by considering all possible group sizes from 2 to 6 members.

Statement (2): Any combination of members is acceptable except those with 1 member.

This statement tells us that all combinations of committee members are acceptable, except for those with only 1 member. It implies that subcommittees can be formed with any number of members greater than or equal to 2.

When we consider each statement alone:

• Statement (1) alone is sufficient to determine the number of subcommittees because it specifies the minimum group size required. By considering all possible group sizes from 2 to 6, we can calculate the number of subcommittees.
• Statement (2) alone is also sufficient because it allows for any combination of members except those with 1 member. This means that subcommittees can be formed with any number of members greater than or equal to 2.

Therefore, each statement alone is sufficient to answer the question asked. The answer is D.

Statement (1): There are 120 ways to form a team of 3 out of all the employees in company C.

This statement tells us the number of ways to form a team of 3, but it doesn't provide information about the total number of employees in company C. However, it implies that there are enough employees in company C to form at least 120 different teams of 3.

Statement (2): All the employees in company C can be divided into two teams of equal employees in 126 ways.

This statement tells us that the employees in company C can be divided into two teams of equal size in 126 different ways. Again, it doesn't provide a direct count of the total number of employees in company C.

When we consider each statement alone, we can see that both statements provide information about the employees and teams, but they don't directly give the total number of employees in company C.

However, when we consider both statements together, we can infer that the total number of employees in company C must be a multiple of 3, as there are 120 ways to form teams of 3. Additionally, we know that the employees can be divided into two equal teams in 126 ways. These two conditions suggest that the number of employees in company C must be a multiple of 6.

Therefore, by combining the information from both statements, we can conclude that EACH statement ALONE is sufficient to answer the question asked. The answer is D.

Statement (I): The number of plants with white flowers is 10.

This statement tells us the number of plants with white flowers, but it doesn't provide any information about the number of black seeds or the relationship between the colors of the seeds and the colors of the flowers. Statement (I) alone is not sufficient to answer the question.

Statement (II): The number of plants with red flowers is 70.

This statement tells us the number of plants with red flowers, but similar to Statement (I), it doesn't provide any information about the number of black seeds or the relationship between the colors of the seeds and the colors of the flowers. Statement (II) alone is not sufficient to answer the question.

When we consider both statements together, we have information about the number of plants with white flowers and the number of plants with red flowers. However, we still don't have any information about the relationship between the colors of the seeds and the colors of the flowers. Without knowing this relationship, we cannot determine the number of black seeds.

Therefore, Statements (I) and (II) together are not sufficient to answer the question asked. The answer is E: Statements (1) and (2) together are NOT sufficient to answer the question asked, and additional data are needed.

To determine the number of ways to select a team from eligible candidates, let’s analyze the information given in each statement.

Statement (1):
The number of eligible candidates is three times the number of slots on the team.

This statement gives us a relationship between the total number of eligible candidates and the team size. However, without knowing the exact number of team slots or eligible candidates, we cannot calculate the total ways of selection. Statement (1) alone is insufficient.**

Statement (2):
60% of the 20 athletes are eligible to play on the four-person university team.

This statement allows us to calculate the number of eligible candidates:
- 60% of 20 athletes = ( 0.6 x20 = 12) eligible candidates.
- The team size is 4.

Knowing the total eligible candidates and team size, we can calculate the number of ways to select the team: ¹²C₄

Statement (2) alone is sufficient.

 Combined Statements:
Since statement (2) already provides all the necessary information, combining both statements is unnecessary.

 Answer:
Statement (2) alone is sufficient.

Statement (1): To guarantee that a red marble is removed, the smallest number of marbles that must be removed from the box is 14.

This statement tells us that in order to ensure the removal of at least one red marble, a minimum of 14 marbles must be removed from the box. However, it doesn't provide any information about the number of yellow or blue marbles in the box. Therefore, Statement (1) alone is not sufficient to answer the question.

Statement (2): To guarantee that a yellow marble is removed, the smallest number of marbles that must be removed from the box is 8.

This statement tells us that in order to ensure the removal of at least one yellow marble, a minimum of 8 marbles must be removed from the box. Similarly to Statement (1), it doesn't provide any information about the number of red or blue marbles in the box. Therefore, Statement (2) alone is not sufficient to answer the question.

When we consider both statements together, we have some additional information. From Statement (1), we know that a minimum of 14 marbles must be removed to guarantee the removal of a red marble. From Statement (2), we know that a minimum of 8 marbles must be removed to guarantee the removal of a yellow marble. This implies that there are at least 14 red marbles and at least 8 yellow marbles in the box. However, we still don't have enough information to determine the exact difference between the number of yellow marbles and red marbles.

Therefore, both statements (1) and (2) together are sufficient to answer the question asked, but neither statement alone is sufficient. The answer is C: BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient.

Statement (1): If two more women were available for selection, exactly 56 different groups of three women could be selected.

From this statement, we know that if two more women were available, there would be exactly 56 different groups of three women that could be selected. However, this statement alone does not provide information about the number of men or how the men and women should be combined to form panels. Therefore, Statement (1) alone is not sufficient to answer the question.

Statement (2): x = y + 1

This statement tells us that the number of women, x, is equal to the number of men plus one, y + 1. This gives us a relationship between the number of women and the number of men available. However, it does not provide information about the total number of women and men or how the panels should be formed. Therefore, Statement (2) alone is not sufficient to answer the question.

When we consider both statements together, we have some additional information. From Statement (1), we know that if two more women were available, exactly 56 different groups of three women could be selected. Combining this with Statement (2), which states that x = y + 1, we can determine that the total number of women available is x = 56 + 2 = 58 and the total number of men available is y = 58 - 1 = 57. With this information, we can calculate the number of different panels that can be formed by choosing three women and two men from the given numbers. The calculation would be the combination of 58 choose 3 (C(58, 3)) multiplied by the combination of 57 choose 2 (C(57, 2)). This will give us the number of different panels that can be formed with the given constraints.

Therefore, both statements (1) and (2) together are sufficient to answer the question asked, but neither statement alone is sufficient. The answer is C: BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient.