A certain company assigns employees to offices in such a way that some of the offices can be empty and...

1. Translate the problem requirements: We need to find the number of ways to assign 3 employees to 2 offices, where offices can be empty and multiple employees can share an office. This means each employee independently chooses one of the two offices.
2. Identify the independent choices: Recognize that each employee makes an independent decision about which office to choose, with no restrictions on the assignments.
3. Apply the multiplication principle: Since each employee has the same number of office choices and these choices are independent, multiply the number of choices for each employee.
4. Calculate the total arrangements: Compute the final result using the multiplication principle.

Execution of Strategic Approach

1. Translate the problem requirements

Let's break down what this problem is asking in everyday language. We have 3 employees and 2 offices. The key conditions are:

• Offices can be empty (no employee has to go to every office)
• Multiple employees can share the same office
• We need to count all possible ways to assign these 3 employees

Think of it this way: each employee walks into the building and has to choose between Office A or Office B. There are no restrictions - they can all crowd into one office if they want, or split up however they like.

Process Skill: TRANSLATE - Converting the assignment language into the core concept that each employee makes an independent choice

2. Identify the independent choices

This is the crucial insight: each employee's choice is completely independent of what the others do.

Let's call our employees Employee 1, Employee 2, and Employee 3, and our offices Office A and Office B.

• Employee 1 can choose: Office A or Office B (2 choices)
• Employee 2 can choose: Office A or Office B (2 choices)
• Employee 3 can choose: Office A or Office B (2 choices)

No matter what Employee 1 chooses, Employee 2 still has both offices available. Same for Employee 3. This independence is what makes the problem solvable using basic counting principles.

3. Apply the multiplication principle

Since each choice is independent, we can use the fundamental counting principle: when you have a sequence of independent choices, you multiply the number of options for each choice.

In plain English:

• First employee has 2 office choices
• Second employee has 2 office choices
• Third employee has 2 office choices
• Total ways = \(2 \times 2 \times 2\)

This gives us: \(2^3 = 8\) total ways

4. Calculate the total arrangements

Let's verify this by listing out all possibilities to make sure our logic is correct:

Employee assignments (Office A, Office B):

1. (1,2,3) - All three in Office A
2. (1,2) (3) - Employees 1,2 in A; Employee 3 in B
3. (1,3) (2) - Employees 1,3 in A; Employee 2 in B
4. (1) (2,3) - Employee 1 in A; Employees 2,3 in B
5. (2,3) (1) - Employees 2,3 in A; Employee 1 in B
6. (2) (1,3) - Employee 2 in A; Employees 1,3 in B
7. (3) (1,2) - Employee 3 in A; Employees 1,2 in B
8. () (1,2,3) - All three in Office B

Counting these up: 8 different arrangements, which confirms our calculation.

Final Answer

The company can assign 3 employees to 2 different offices in 8 different ways.

The answer is D. 8

Common Faltering Points

Errors while devising the approach

1. Misinterpreting this as a permutation instead of assignment problem

Students often confuse this with arranging employees in specific positions within offices, leading them to consider order within offices as important. They might think "Employee 1 in Office A, Employee 2 in Office A" is different from "Employee 2 in Office A, Employee 1 in Office A" and try to use permutation formulas like \(\mathrm{P}(3,2)\) or factorial calculations.

2. Overlooking the constraint that offices can be empty

Many students assume that each office must have at least one employee assigned to it, similar to distribution problems where every container must be used. This leads them to exclude valid scenarios where all employees could be assigned to just one office, significantly undercounting the total possibilities.

3. Failing to recognize the independence of employee choices

Students may think that once one employee chooses an office, it affects the choices available to other employees. They might approach this as a sequential selection problem with dependencies rather than recognizing that each employee's choice is completely independent of others.

Errors while executing the approach

1. Calculation errors with exponents

Even when students correctly identify this as \(2^3\), they might make basic arithmetic mistakes, calculating \(2^3\) as 6 (confusing it with \(2 \times 3\)) or 9 (confusing it with \(3^2\)) instead of the correct value of 8.

2. Incomplete enumeration when listing cases

When attempting to verify by listing all possibilities, students commonly miss cases where all employees go to the same office, particularly the case where everyone goes to Office B. They tend to focus on mixed distributions and forget the extreme cases.

Errors while selecting the answer

No likely faltering points - The calculation directly gives 8, which matches answer choice D exactly, leaving little room for selection errors.