Test: Combinations - GMAT MCQ

Statement (1): Each team in league X played each of the other teams at least once.

This statement tells us that each team played against every other team at least once. However, it does not provide information about the number of games played between any two specific teams or the overall schedule of the matches. Therefore, statement (1) alone is not sufficient to determine the total number of games played.

Statement (2): No team in league X played more than 7 games.

This statement tells us that the maximum number of games played by any team is 7. However, it does not provide information about the minimum or average number of games played by each team or the total number of games played by all teams combined. Therefore, statement (2) alone is not sufficient to determine the total number of games played.

When we consider both statements together:

From statement (1), we know that each team played against every other team at least once.

From statement (2), we know that no team played more than 7 games.

However, even when considering both statements together, we do not have enough information to determine the exact number of games played by the 6 teams. We do not know if there were any additional games played beyond the minimum required for each team to play against every other team at least once.

Therefore, statements (1) and (2) together are not sufficient to determine the total number of games played. The answer is (E) Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data are needed.

Statement (1): A triangle is formed by joining 3 distinct points in the plane.

This statement tells us that a triangle is formed by selecting 3 distinct points among the given 8 points. However, it does not provide us with information about the arrangement or positioning of these points. Therefore, statement (1) alone is not sufficient to determine the number of triangles that can be formed.

Statement (2): Out of 8 given points, three are collinear.

This statement tells us that three of the given 8 points lie on the same line. If three points are collinear, they cannot form a triangle since a triangle requires three non-collinear points. Therefore, if three points are collinear among the 8 given points, no triangle can be formed.

From statement (2) alone, we can conclude that the number of triangles that can be formed is zero since there are three collinear points.

Therefore, statement (2) alone is sufficient to answer the question. The answer is (B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked.

Statement (1): Players must match at least two numbers with the machine to win.

This statement tells us that in order to win, a player must match at least two numbers. However, it does not provide us with any information about the total number of digits or the number of digits chosen by the player. Therefore, statement (1) alone is not sufficient to determine the probability of winning.

Statement (2): X = 4

This statement tells us that the player selects 4 distinct single-digit numbers. However, it does not provide us with information about the requirements for winning the lottery. Therefore, statement (2) alone is not sufficient to determine the probability of winning.

When we consider both statements together:

From statement (2), we know that X = 4, which means the player selects 4 distinct single-digit numbers.

From statement (1), we know that the player must match at least two numbers to win.

To calculate the probability of winning, we need to consider the total number of possible outcomes and the number of favorable outcomes.

The total number of possible outcomes can be calculated by selecting 4 distinct single-digit numbers out of 10 digits (0 to 9) without considering the order. This can be expressed as C(10, 4) = 10! / [4! * (10 - 4)!] = 210.

Now, let's consider the number of favorable outcomes, which is the number of ways to select at least two matching numbers. Since the player must select 4 distinct single-digit numbers, the only way to have at least two matching numbers is if the player selects two pairs of matching numbers. The number of ways to choose two pairs of matching numbers is C(5, 2) = 5! / [2! * (5 - 2)!] = 10.

Therefore, the probability of winning the lottery is 10/210 = 1/21.

By considering both statements together, we were able to determine the probability of winning the lottery. Hence, 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): The number of boys is 20% less than the number of girls in the class.

Let's assume the number of girls in the class is G. According to statement (1), the number of boys would be 0.8G (20% less than the number of girls).

To find the number of ways the teacher can choose two boys, we need to determine the combination of 2 out of the number of boys.

The number of ways to choose 2 boys out of 0.8G boys is given by the expression C(0.8G, 2) = (0.8G)! / [2!(0.8G - 2)!].

Statement (1) provides us with the necessary information to calculate the number of ways to choose two boys.

Statement (2): The number of girls is 3 more than the number of boys in the class.

Let's assume the number of boys in the class is B. According to statement (2), the number of girls would be B + 3.

However, statement (2) alone does not provide us with any specific information about the number of boys or girls in the class. Therefore, we cannot determine the number of ways to choose two boys solely based on statement (2) alone.

When we consider both statements together:

From statement (1), we have B = 0.8G.

From statement (2), we have G = B + 3.

Substituting the value of B from statement (1) into statement (2), we get G = 0.8G + 3.

Simplifying this equation, 0.2G = 3.

Dividing both sides by 0.2, we find G = 15.

Substituting G = 15 into B = 0.8G from statement (1), we get B = 0.8 * 15 = 12.

Therefore, there are 12 boys and 15 girls in the class.

The number of ways to choose 2 boys out of 12 boys is C(12, 2) = 66.

Therefore, both statements together are sufficient to determine the number of ways the teacher can choose two boys. Hence, 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): The number of different possible sequences of n-digit numbers when a dice is rolled n times is 7776.

If we roll a fair six-sided dice n times, the total number of possible outcomes would be 6ⁿ since each roll has 6 possible outcomes. Therefore, statement (1) tells us that 6ⁿ = 7776.

By taking the logarithm of both sides, we can determine the value of n. log(6^n) = log(7776), which simplifies to n log(6) = log(7776).

We can calculate log(7776) ≈ 3.89 and log(6) ≈ 0.78. Dividing both sides of the equation by log(6), we have n ≈ 3.89 / 0.78 ≈ 5.

So, statement (1) alone is sufficient to determine that n is approximately 5.

Statement (2): If the dice has been rolled 3 times fewer, the probability of getting a 6 on every roll would have been 1/36.

Let's denote the original number of rolls as n. According to statement (2), if we roll the dice n - 3 times, the probability of getting a 6 on every roll would be 1/36.

The probability of getting a 6 on a single roll of a fair six-sided dice is 1/6. Therefore, the probability of getting a 6 on every roll in n - 3 rolls would be (1/6)^{(n - 3)}.

According to statement (2), (1/6)^{(n - 3)} = 1/36.

To solve this equation, we can raise both sides to the power of -1/2, resulting in (1/6)^{(n - 3)} = (1/36)^(-1/2).

Simplifying further, (1/6)^{(n - 3)} = 6.

Taking the logarithm of both sides, we get (n - 3) log(1/6) = log(6).

Here, log(1/6) ≈ -0.78. Dividing both sides by log(1/6), we have n - 3 ≈ log(6) / (-0.78).

Approximately, n - 3 ≈ -7.69. Adding 3 to both sides, we get n ≈ -4.69.

The value of n cannot be negative, so this solution is not valid. Hence, statement (2) alone does not provide a valid solution for the value of n.

Considering both statements together, we know from statement (1) that n ≈ 5. Since statement (2) does not contradict this value, it is consistent with the value of n obtained from statement (1).

Therefore, each statement alone is sufficient to answer the question asked. The answer is (D) EACH statement ALONE is sufficient to answer the question asked.

Statement (1): Ethel answered 10 more questions correctly on the first half of the test than on the second half.

This statement provides information about Ethel's performance on different halves of the test. However, it doesn't give us any specific information about the number of questions she answered correctly overall. Therefore, statement (1) alone is not sufficient to determine whether Ethel passed the test.

Statement (2): Ethel answered more than half of the questions on the test correctly.

This statement tells us that Ethel answered more than half of the questions correctly. However, it doesn't provide us with the exact number of questions or the total score. Therefore, statement (2) alone is not sufficient to determine whether Ethel passed the test.

When we consider both statements together, we can deduce the following:

If Ethel answered 10 more questions correctly on the first half than on the second half, it implies that she answered at least 10 questions correctly on the first half. Let's say she answered x questions correctly on the second half. In that case, she answered x + 10 questions correctly on the first half. Since the total number of questions is 50, Ethel answered a total of (x + 10) + x = 2x + 10 questions correctly.

Now, statement (2) tells us that Ethel answered more than half of the questions correctly. So, 2x + 10 > 50/2, which simplifies to 2x + 10 > 25.

Subtracting 10 from both sides of the inequality gives us 2x > 15. Dividing both sides by 2, we get x > 7.5.

Since x represents the number of questions Ethel answered correctly on the second half, it must be a whole number. Therefore, x can be at least 8.

If x is 8, then Ethel answered (8 + 10) + 8 = 26 questions correctly in total, resulting in a score of 52 points (26 * 2). This score is below the passing score of 60. Therefore, Ethel did not pass the test.

If x is greater than 8, the total score would be even lower, further confirming that Ethel did not pass the test.

Therefore, both statements together are sufficient to determine that Ethel did not pass the test. Hence, 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): In total, six scholarships will be granted.

From statement (1), we know that there will be a total of six scholarships granted. However, we don't have any information about how these scholarships will be distributed among the different levels (e.g., $10,000, $5,000, $1,000). Therefore, statement (1) alone is not sufficient to determine the number of different ways the scholarships can be doled out.

Statement (2): An equal number of scholarships will be granted at each scholarship level.

From statement (2), we know that an equal number of scholarships will be granted at each scholarship level. However, we don't have any specific information about the number of scholarships at each level or the total number of scholarships. Therefore, statement (2) alone is not sufficient to determine the number of different ways the scholarships can be doled out.

Combining both statements, we know that a total of six scholarships will be granted, and an equal number of scholarships will be granted at each level. However, we still don't have enough information to determine the number of different ways the scholarships can be doled out among the pool of 10 applicants. We don't know the specific distribution of scholarships among the different levels or the number of scholarships at each level.

Therefore, both statements together are sufficient to answer the question asked, but neither statement alone is sufficient. Thus, 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): The number of eligible candidates is three times as great as the number of slots on the team.

Let's denote the number of eligible candidates as N. According to statement (1), N = 3 × 4 = 12. In this case, there are 12 eligible candidates competing for 4 slots on the team. The number of ways to select a 4-person team from 12 eligible candidates can be calculated using the combination formula: C(12, 4) = 12! / (4! × (12-4)!) = 495.

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

Let's denote the number of eligible athletes as E. According to statement (2), E = 60% of 20 = 0.6 × 20 = 12. Here, we know the number of eligible athletes but don't have any information about the total number of candidates. We cannot determine the number of ways to select a 4-person team solely based on statement (2) since we don't know the total number of candidates.

Combining both statements, we have N = 12 and E = 12. Both statements provide the same information, so together they are redundant. We can use statement (1) alone to determine the number of ways to select a 4-person university team, which is 495. Therefore, the answer is D: EACH statement ALONE is sufficient to answer the question asked.

To determine if x > y, let's analyze the given statements:

Statement (1): x + y > 0
This statement alone tells us that the sum of x and y is greater than zero, but it doesn't provide any specific information about the relationship between x and y individually. Therefore, statement (1) alone is not sufficient to answer the question.

Statement (2): y² > x²
This statement tells us that the square of y is greater than the square of x. However, this doesn't necessarily mean that y is greater than x because the squares of both numbers could be equal while the numbers themselves have different signs. For example, if x = -2 and y = 2, the condition in statement (2) would be satisfied, but x would still be greater than y. Therefore, statement (2) alone is not sufficient to answer the question.

Considering both statements together, we still don't have enough information to determine if x > y. Statement (1) provides information about the sum of x and y, while statement (2) provides information about the squares of x and y. However, the relationship between x and y is not explicitly established.

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

Statement (1): x² < y²
This statement tells us that the square of x is less than the square of y. Since both sides of the inequality are squared, we can take the square root of both sides while preserving the inequality. Taking the square root, we get |x| < |y|. This means that the absolute value of x is less than the absolute value of y. However, this does not provide enough information to determine the relationship between x and y individually. For example, x could be -3 and y could be -2, in which case x is less than y. Conversely, x could be -2 and y could be -3, in which case y is less than x. Therefore, statement (1) alone is not sufficient to answer the question.

Statement (2): y < 0
This statement tells us that y is less than zero, or in other words, y is negative. However, this does not provide any information about the relationship between x and y. For example, x could be positive or negative, and the comparison between x and y would still be unknown. Therefore, statement (2) alone is not sufficient to answer the question.

Considering both statements together, we can infer the following:
From statement (1), |x| < |y|
From statement (2), y < 0

Since y is negative, it means that |y| = -y. Therefore, from statement (1), we have |x| < -y. Since the absolute value of a number is always non-negative, |x| < -y is only possible if x is negative. So, we have x < 0.

Based on the information obtained from both statements, we can conclude that x is negative (x < 0), but we cannot determine the relationship between x and y in terms of which one is greater. Therefore, both statements together are sufficient to answer the question asked, but neither statement alone is sufficient.

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