Test: Combinations - GMAT MCQ

Let's analyze each statement and see if they provide enough information to determine the value of R.

Statement (1): R² + 2R - 8 = 0

To solve this quadratic equation, we can factorize it as follows:
(R + 4)(R - 2) = 0

Setting each factor equal to zero, we find two possible solutions for R: R = -4 or R = 2.

However, we are looking for a positive number of employees on the party planning committee, so R = -4 is not a valid solution. Therefore, the only valid solution is R = 2.

Statement (1) alone is sufficient to determine the value of R.

Statement (2): There are 15 different ways to choose R employees to be on the party planning committee.

This statement alone does not provide any specific information about the value of R. It only tells us the number of different ways to choose R employees, but it doesn't give us the actual value of R.

Statement (2) alone is not sufficient to determine the value of R.

Combining both statements:
From Statement (1), we know that R = 2.

From Statement (2), we know that there are 15 different ways to choose 2 employees for the party planning committee.

Together, the statements are consistent and provide a unique value for R.

Therefore, both statements together are sufficient to determine the value of R.

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

Statement (1): M < 11
This statement provides information about the maximum number of male managers, which is less than 11. However, it does not provide any information about the number of female executives or the total number of people present at the meeting. Therefore, statement (1) alone is not sufficient to answer the question.

Statement (2): M(M + 2N) = 65
This statement provides an equation relating the number of male managers (M), the number of female executives (N), and the total number of shaking hands. However, we still need to determine the values of M and N to calculate the total number of shaking hands or the number of shaking hands among the female executives only. Therefore, statement (2) alone is not sufficient to answer the question.

Now, let's consider both statements together:

From statement (2), we have M(M + 2N) = 65.

Since the total number of shaking hands is the sum of the shaking hands among the female executives and the shaking hands between male managers and female executives, we can express it as follows:
Total shaking hands = Shaking hands among female executives + Shaking hands between male managers and female executives.

The shaking hands among the female executives can be calculated by selecting 2 out of N people, which is given by the combination formula: N(N-1)/2.

The shaking hands between male managers and female executives can be calculated by multiplying the number of male managers (M) by the number of female executives (N).

Therefore, we can write the equation:
M(M + 2N) = N(N-1)/2 + MN

Simplifying this equation, we get:
M² + 2MN = N(N-1)/2 + MN
M² + MN = N(N-1)/2

We can see that we have one equation with two variables, M and N. Without additional information, we cannot determine the values of M and N uniquely. Therefore, statements (1) and (2) together are not sufficient to answer the question.

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

To determine the number of different teams the sales manager can select, let's analyze each statement separately.

Statement (1): The team will comprise either 1/8 or 1/6 of the total number of salespeople, depending on the size of the team.

Let's assume there are N salespeople in total.

If the team comprises 1/8 of the total number of salespeople, then the number of team members will be N/8.
If the team comprises 1/6 of the total number of salespeople, then the number of team members will be N/6.

However, this statement alone does not provide us with the exact value of N or whether it is divisible evenly by 6 or 8. Therefore, statement (1) alone is not sufficient to answer the question.

Statement (2): It is suspected that 20 salespeople will NOT be selected to be part of this team.

This statement doesn't provide us with any information about the size of the team or the total number of salespeople. It only indicates that 20 salespeople will not be selected. Without further information, we cannot determine the number of different teams. Therefore, statement (2) alone is not sufficient to answer the question.

Since neither statement alone is sufficient, let's consider both statements together.

Combining both statements, we know that the team size can be either N/8 or N/6, and 20 salespeople will not be selected. However, we still don't have the exact value of N or any information about its divisibility by 6 or 8. Therefore, even when considering both statements together, we still can't determine the number of different teams.

Hence, the answer is (A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked.

Let's analyze each statement individually:

Statement (1) alone: We know that more than 5 individuals moved out of the room. However, we don't have any information about the number of sibling pairs among those who moved out or those who remained. It's possible that all the individuals who moved out were from different sibling pairs, leaving all 10 sibling pairs still in the room. Alternatively, it's also possible that some or all of the individuals who moved out belonged to sibling pairs, reducing the number of pairs remaining. Without knowing the distribution of sibling pairs among those who moved out and those who remained, we cannot determine if the number of sibling pairs remaining is greater than 4. Statement (1) alone is not sufficient.

Statement (2) alone: We know that fewer than 12 individuals moved out of the room. Similar to Statement (1), we lack information about the distribution of sibling pairs among those who moved out and those who remained. Therefore, we cannot determine if the number of sibling pairs remaining is greater than 4 based solely on Statement (2) alone. It is not sufficient.

Considering both statements together: When we consider both statements together, we still don't have any information about the distribution of sibling pairs among those who moved out and those who remained. Therefore, even when considering both statements, we cannot determine if the number of sibling pairs remaining is greater than 4. Both statements together are not sufficient.

Since we cannot determine the answer to the question with the given information, the correct answer is (E) Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data are needed.

Given:
From a group of 6 employees, k employees are chosen to be on the party-planning committee. If k is a positive integer, we need to find the value of k.

We have two statements to consider:

Statement (1): k is a prime number.
Statement (2): There are 15 different ways to create the party-planning committee consisting of k employees.

Let's analyze each statement separately:

Statement (1) alone:
If k is a prime number, we know it is greater than 1 and can only be divided evenly by 1 and itself. However, this information does not provide any direct information about the value of k. It could be any prime number greater than 1. Therefore, statement (1) alone is not sufficient to answer the question.

Statement (2) alone:
Statement (2) tells us that there are 15 different ways to create the party-planning committee consisting of k employees. We know that the total number of employees is 6, and we are selecting k employees. Since there are 15 different ways to create the committee, we can set up an equation to solve for k:

C(6, k) = 15

Here, C(n, r) represents the combination function, which calculates the number of ways to choose r items from a set of n items.

Unfortunately, this equation is not enough to determine the value of k uniquely. We need additional information to solve it. Therefore, statement (2) alone is not sufficient to answer the question.

Considering both statements together:
When we combine the information from both statements, we have some constraints. We know that k must be a prime number and that there are 15 different ways to create the committee. By combining these constraints, we can determine the possible values of k.

If we list the prime numbers greater than 1 up to 6, we have: 2, 3, 5.

Now, we can calculate C(6, k) for each of these values:

C(6, 2) = 15
C(6, 3) = 20
C(6, 5) = 6

From these calculations, we see that only C(6, 2) = 15, satisfies the condition stated in statement (2). Therefore, the value of k is 2.

Hence, both statements together are sufficient to answer the question, but neither statement alone is sufficient.

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

To solve this problem, let's analyze each statement individually and then evaluate their combined information.

Statement (1): The number of different permutations of the numbers in the list is 12.

From the given list, {a, b, 1, 2}, we need to determine the mode.
The mode is the value that appears most frequently in the list.
However, Statement (1) provides information about the number of different permutations, which is not directly related to finding the mode.
The statement alone does not provide any information about the specific values of a and b or their frequencies in the list.
Therefore, Statement (1) alone is not sufficient to answer the question.

Statement (2): A four-digit number 21ab is divisible by 9.

For a number to be divisible by 9, the sum of its digits must be divisible by 9.
The four-digit number 21ab can be expressed as 2100 + 10a + b.
The sum of its digits is (2 + 1 + 0 + 0) + (a) + (b).
For the entire expression to be divisible by 9, the sum of its digits must be divisible by 9.

However, this statement does not provide any direct information about the mode or the frequencies of the numbers in the list.
It only provides a condition for divisibility by 9.
Therefore, Statement (2) alone is not sufficient to answer the question.

Combining the statements:

By combining the statements, we still don't have enough information to determine the mode.
The first statement provides information about permutations, while the second statement provides a condition for divisibility by 9.
However, neither statement gives us direct information about the mode or the frequencies of the numbers in the list.
Therefore, the statements together are still not sufficient to answer the question.

Based on the analysis of both statements individually and together, we can conclude that the answer is (E) Statements (1) and (2) together are not sufficient to answer the question asked, and additional data are needed.

The question asks for the number of different committees of y people that can be chosen from a group of x people, given that x > y.

Statement (1) alone: The number of different committees of x-y people that can be chosen from a group of x people is 3,060.

This statement provides information about the number of committees of x-y people, but it does not directly provide information about the number of committees of y people. Therefore, statement (1) alone is not sufficient to answer the question.

Statement (2) alone: The number of different ways to arrange x-y people in a line is 24.

This statement provides information about arranging x-y people in a line, but it does not provide information about committees or the number of people in the committees. Therefore, statement (2) alone is not sufficient to answer the question.

Considering both statements together:

Let's assume the number of people in the group is x and the number of people in the committee is y.

Statement (1) tells us that the number of committees of x-y people that can be chosen from a group of x people is 3,060. This can be represented as xC(x-y) = 3,060.

Statement (2) does not provide direct information about committees or the number of people in the committees, so it does not directly contribute to solving the problem.

By combining the two statements, we have:

xC(x-y) = 3,060

Since x > y, we know that x-y is positive.

We need to find the value of y, which represents the number of people in the committee.

Since we have only one equation and two variables (x and y), we cannot uniquely determine the values of x and y. Therefore, statements (1) and (2) together are not sufficient to answer the question.

Based on the analysis above, the answer is (A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked.

To solve this problem, let's analyze each statement and then combine them to determine the number of seating arrangements.

Statement (1): N = 5

This statement tells us that there are 5 couples in the group. Since each couple insists on sitting together, we can consider each couple as a single unit. Therefore, we have a total of 5 units to arrange in a row.

Statement (2): The group will all sit next to one another, starting with the first seat in the row.

This statement indicates that all members of the group will sit together in a contiguous manner, starting from the first seat in the row. This means that the 5 units representing the couples will be treated as a single block.

Combining the statements:

Based on statement (2), we know that the 5 units representing the couples will form a single block. Now, we need to determine the number of arrangements within this block.

Since the row has more than 2N seats, we can assume that there are enough seats to accommodate the block of couples. Within this block, the 5 units can be arranged in 5! = 120 ways.

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

We have four men and three women, and they need to be seated in a row of seven chairs. Since the committee has one male captain and one female captain, we know that these two individuals will occupy the first and seventh chairs. The remaining five members will fill the chairs in between.

Statement (1) tells us that no woman sits next to another woman. This means that the three women cannot be seated consecutively. Let's consider the possibilities:

• If we have three consecutive chairs occupied by women, the only option is to have men sitting on both sides. For example: M W W W M M M. In this case, at least one man sits next to another man.
• If we have two consecutive chairs occupied by women, there will be two options: W W M W M M M or M W W M W M M. In both cases, at least one man sits next to another man.
• If we have one chair occupied by a woman, there will be three options: W M W M W M M, M W M W M W M, or M M W M W M W. In all three cases, at least one man sits next to another man.
• If there are no consecutive chairs occupied by women, there will be several possibilities. However, in all these cases, there will still be at least one man sitting next to another man.

Therefore, statement (1) alone is sufficient to conclude that at least one man sits next to another man.

Now let's consider statement (2). It states that the captains sit in the first and seventh chairs. Based on this information, we know the specific positions of two individuals. However, it doesn't provide any information about the arrangement of the remaining committee members. Therefore, statement (2) alone is not sufficient to answer the question.

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

Statement (1): If one less potential juror had been rejected, it would be possible to create 13 different juries.

This statement suggests that if one potential juror had not been rejected, there would be 13 different possible juries. This means that the total number of potential jurors initially is 13 (12 + 1), and the defense council and prosecuting attorney rejected 1 potential juror. However, this statement alone does not provide enough information to determine the total number of potential jurors (n) or the number of potential jurors rejected (m).

Statement (2): n = m + 12.

This statement indicates that the total number of potential jurors (n) is equal to the number of potential jurors rejected (m) plus 12. This implies that the number of potential jurors initially is 12 more than the number rejected. However, this statement alone does not provide the exact values of n or m.

Combined, we can use both statements to solve the problem:

From statement (2), we have n = m + 12.

From statement (1), we know that if one less potential juror had been rejected, there would be 13 different juries. This means that the total number of potential jurors initially is 13 (12 + 1).

Combining these two statements, we have:

13 = m + 12.

Simplifying the equation, we find m = 1.

Now we can substitute the value of m back into statement (2) to find n:

n = 1 + 12 = 13.

Therefore, there are 13 potential jurors initially, and 1 potential juror was rejected.

To calculate the number of different possible juries that can be picked from the remaining potential jurors, we can use the combination formula:

C(n, 12) = C(13, 12) = 13.

So, there are 13 different possible juries that can be picked from the remaining potential jurors.

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