Read the following details carefully and answer the questions that follow.

Nine consultants are assigned to three different departments: Department Alpha, Department Beta, and Department Gamma. Not less than two consultants and not more than four consultants work in any department. David works only with Emily, but not in department Beta. Frank works in department Beta but not with Ivy. Jack, Karen, and Leo work in the same department. Karen works with Ivy but not in department Alpha. Grace and Henry are two of the consultants.

Which of the following statements is/are correct?
I. Grace works with Frank
II. Jack and Henry work in the same department
III. Emily works in department Alpha

Correct Answer: Both statements I and III are correct

Step-by-Step Explanation:

Let's analyze the given conditions step-by-step to determine the department assignment for each of the 9 consultants: David, Emily, Frank, Grace, Henry, Ivy, Jack, Karen, and Leo across three departments (Alpha, Beta, Gamma).

Condition 1: Department Capacity Constraints

There are 9 consultants in total across 3 departments (Alpha, Beta, Gamma). Each department has between 2 and 4 consultants (2 ≤ capacity ≤ 4).

Condition 2: David and Emily's placement

"David works only with Emily, but not in department Beta."
This means the department where David works contains exactly 2 consultants: David and Emily.
Since David does not work in Beta, David and Emily must work in either Department Alpha or Department Gamma.

Condition 3: Jack, Karen, Leo, and Ivy's placement

1. "Jack, Karen, and Leo work in the same department." -> This forms a group of 3 consultants: {Jack, Karen, Leo}.
2. "Karen works with Ivy but not in department Alpha." -> Ivy must be in the same department as Karen, Jack, and Leo.
3. This group of 4 consultants: {Jack, Karen, Leo, Ivy} must be together in one department.
4. Since Karen is not in department Alpha, and David & Emily occupy a department with only 2 people, the group {Jack, Karen, Leo, Ivy} (which has 4 people) cannot be in Alpha or David's department.
5. Therefore, {Jack, Karen, Leo, Ivy} must be assigned to either Beta or Gamma.

Condition 4: Frank's placement

"Frank works in department Beta but not with Ivy."
Since Frank works in department Beta and does NOT work with Ivy, the 4-person group {Jack, Karen, Leo, Ivy} cannot be in department Beta. Thus, {Jack, Karen, Leo, Ivy} must be in Department Gamma.

Determining the exact department breakdown:

1. Department Gamma (4 consultants): Jack, Karen, Leo, Ivy.
2. Now we have 5 consultants left: David, Emily, Frank, Grace, Henry.
3. We know David and Emily work alone together in a department of size 2. Since Gamma has 4 people and Beta has Frank, David & Emily must be in Department Alpha (size = 2).
4. Department Alpha (2 consultants): David, Emily.
5. The remaining consultants must be in Department Beta: Frank, Grace, Henry (size = 3).

Final Department Assignments:

• Department Alpha (2): David, Emily
• Department Beta (3): Frank, Grace, Henry
• Department Gamma (4): Jack, Karen, Leo, Ivy

Evaluating the Statements:

• Statement I: Grace works with Frank.
From our arrangement, Grace and Frank are both in Department Beta. (Correct)

• Statement II: Jack and Henry work in the same department.
Jack is in Department Gamma and Henry is in Department Beta. (Incorrect)

• Statement III: Emily works in department Alpha.
Emily is indeed in Department Alpha. (Correct)

Therefore, statements I and III are correct.