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 true?

Correct Answer: All are correct.

Let's analyze the given statements step-by-step to determine the assignment of all nine consultants (David, Emily, Frank, Grace, Henry, Ivy, Jack, Karen, Leo) into the three departments: Department Alpha, Department Beta, and Department Gamma.

Step 1: Understand the Constraints

1. Total consultants = 9.

2. Departments = Alpha, Beta, Gamma.

3. Each department has between 2 and 4 consultants (inclusive): $2\le \text{count}\le 4$.

Step 2: Analyze Consultant Groups & Departments

• David and Emily: "David works only with Emily, but not in department Beta."
This means David and Emily are the only two consultants in their department. Since David cannot be in Beta, David and Emily must be together in either Department Alpha or Department Gamma, forming a complete department of size 2.

• Jack, Karen, Leo, and Ivy: "Jack, Karen, and Leo work in the same department." "Karen works with Ivy but not in department Alpha."
This implies Jack, Karen, Leo, and Ivy all work together in the same department. Thus, this group consists of 4 consultants: {Jack, Karen, Leo, Ivy}.
Since Karen does not work in Department Alpha, and David + Emily take up a whole department of size 2 (which leaves no room for 4 people), the group {Jack, Karen, Leo, Ivy} must be assigned to Department Gamma or Department Beta.

• Frank, Grace, Henry, and Ivy's placement: "Frank works in department Beta but not with Ivy."
Since Frank is in Department Beta and Ivy is NOT in Department Beta, the 4-person group {Jack, Karen, Leo, Ivy} cannot be in Department Beta. Therefore, {Jack, Karen, Leo, Ivy} MUST be in Department Gamma.

Since Department Gamma has 4 consultants {Jack, Karen, Leo, Ivy}:
- The 2-person group {David, Emily} cannot be in Department Gamma. Thus, {David, Emily} MUST be in Department Alpha.

Now, let's look at Department Beta:
- Total consultants = 9.
- Department Alpha has 2 consultants: {David, Emily}.
- Department Gamma has 4 consultants: {Jack, Karen, Leo, Ivy}.
- Remaining consultants for Department Beta = 9 - 2 - 4 = 3 consultants.
- We know Frank is in Department Beta.
- The remaining two consultants are Grace and Henry. Therefore, Department Beta consists of 3 consultants: {Frank, Grace, Henry}.

Step 3: Summary of Assignments

• Department Alpha (2 consultants): David, Emily

• Department Beta (3 consultants): Frank, Grace, Henry

• Department Gamma (4 consultants): Jack, Karen, Leo, Ivy

Step 4: Verify the Given Statements

1. Leo works in department Gamma: True (Leo is in Gamma along with Jack, Karen, and Ivy).

2. Grace and Henry work in the same department: True (Both are in Beta along with Frank).

3. Frank works in department Beta: True (Given directly in the problem statement).

4. Four consultants work in department Gamma: True (Jack, Karen, Leo, and Ivy make 4 consultants).

Since all individual options are correct, the statement "All are correct." is true.