Understanding the Company Department Distribution

This question asks us to determine the distribution of nine people (G, H, I, J, K, L, M, N, O) across three departments (Marketing, Finance, and Admin) and then identify the department with the maximum number of people. We are given several clues and constraints: a minimum of 2 and a maximum of 4 people per department, and the total number of people is 9.

Analyzing the Clues

Let's break down the information provided:

• Total people: 9
• Departments: Marketing, Finance, Admin
• Constraint: 2 <= people per department <= 4
• I is not in Admin.
• G is in Finance.
• M is not in Marketing and not with G (Finance). This means M is in Admin.
• N is not in Marketing.
• O is with G. So, O is in Finance.
• H and K work together but not in Marketing. They are in Finance or Admin.
• J neither works in Finance nor with M (Admin). This means J is in Marketing.
• N is not with K.
• L is working with only one person. This means L is in a department with exactly 2 people.

Step-by-Step Deduction

Let's place people based on the definite clues:

• G is in Finance.
• O is with G, so O is in Finance. (Finance: G, O)
• M is not in Marketing and not with G (Finance). Thus, M must be in Admin. (Admin: M)
• J is not in Finance and not with M (Admin). Thus, J must be in Marketing. (Marketing: J)

Currently, we have:

• Finance: G, O (2 people)
• Marketing: J (1 person)
• Admin: M (1 person)

We still need to place H, I, K, L, and N (5 people). The department sizes must be between 2 and 4. The current distribution violates the minimum size rule for Marketing and Admin. The total size is 2+1+1 = 4. We need to add 5 people.

L is with only one person, meaning L is in a department of size 2. Let's consider where L could be placed:

Possibility 1: L is in Marketing.

If L is in Marketing, Marketing has J and L (size 2). This satisfies L's constraint. Remaining people: H, I, K, N (4 people). Current departments: Finance (G, O - size 2), Marketing (J, L - size 2), Admin (M - size 1). The department sizes must sum to 9. Current sum is 2+2+1=5. We need to add 4 people (H, I, K, N) to Finance and Admin, making their total size 4+1+4 = 9. Possible sizes for Finance and Admin must sum to 7 (9 - 2 in Marketing) and respect min/max rules (2-4). The possible combinations summing to 7 are (3, 4) or (4, 3).

H and K are together (size 2) and not in Marketing. So H and K must go to either Finance or Admin.

• Subcase 1a: H and K go to Finance. Finance: G, O, H, K (size 4). Remaining people: I, N (2 people). They must go to Admin. Admin: M, I, N (size 3). Distribution: Marketing (J, L - 2), Finance (G, O, H, K - 4), Admin (M, I, N - 3). Let's check constraints: - Min 2, Max 4: Yes (2, 4, 3). - I not in Admin: No, I is in Admin in this case. This distribution is invalid.
• Subcase 1b: H and K go to Admin. Admin: M, H, K (size 3). Remaining people: I, N (2 people). They must go to Finance. Finance: G, O, I, N (size 4). Distribution: Marketing (J, L - 2), Finance (G, O, I, N - 4), Admin (M, H, K - 3). Let's check constraints: - Min 2, Max 4: Yes (2, 4, 3). - I not in Admin: Yes, I is in Finance. - G in Finance: Yes. - M not in Marketing and not with G: Yes, M is in Admin. - N not in Marketing: Yes, N is in Finance. - O with G: Yes, O is in Finance. - H and K together, not in Marketing: Yes, H and K are in Admin. - J neither works in Finance nor with M: Yes, J is in Marketing, M is in Admin. - N is not with K: Yes, N is in Finance, K is in Admin. - L working with only one person: Yes, L is with J in Marketing (size 2). All constraints are satisfied by this distribution.

Since Subcase 1b works, we have found the correct distribution. We don't need to explore other main possibilities (like L being in Finance or Admin) as only one valid distribution is possible from the given clues.

Final Department Distribution

The distribution is:

Finding the Department with Maximum People

Comparing the number of people in each department:

• Marketing: 2
• Finance: 4
• Admin: 3

The maximum number of people working in a department is 4, which is in the Finance department.

Conclusion

The department with the maximum number of people working is Finance.