From a committee of 25 legislators, n members formed a task force. Each member of the task force sent a...

1. Translate the problem requirements: Clarify what "reports" and "letters" mean in this context, identify who sends what to whom, and understand that we need the total count of both communications combined.
2. Identify the communication flow: Determine the two distinct communication processes - reports from task force to non-task force members, and letters sent in response.
3. Count each type of communication: Calculate the number of reports sent and the number of letters sent separately using the given constraints.
4. Combine and match to answer choices: Add the two counts together and identify which answer choice represents this total.

Execution of Strategic Approach

1. Translate the problem requirements

Let's start by understanding what's happening in everyday terms. We have a committee of 25 legislators total. From this group, n members form a smaller task force.

Now, here's what happens next:

• Each task force member sends a report to every legislator who is NOT on the task force
• In response to each report received, a letter is sent back

We need to find the total number of both reports AND letters sent combined.

Let me clarify the key terms:

• "Reports" = communications sent FROM task force members TO non-task force members
• "Letters" = communications sent back as responses to the reports
• We want the sum of reports + letters

Process Skill: TRANSLATE - Converting the problem's business language into clear mathematical relationships

2. Identify the communication flow

Let's think through this step by step using plain English logic:

First, let's figure out how many people are in each group:

• Task force members: n people
• Non-task force members: Since there are 25 total legislators and n are on the task force, there must be \(\mathrm{(25 - n)}\) people NOT on the task force

Now for the communication flow:

• Direction 1: Task force members → Non-task force members (these are the "reports")
• Direction 2: Non-task force members → Task force members (these are the "letters" sent in response)

Notice that these are two separate, distinct types of communication happening.

3. Count each type of communication

Let's count the reports first:
Each of the n task force members sends a report to each of the \(\mathrm{(25 - n)}\) non-task force members.

Using concrete thinking: If I'm one task force member, I send reports to all \(\mathrm{(25 - n)}\) non-task force people. Since there are n task force members total, and each does this same thing, the total number of reports is:
Number of reports = \(\mathrm{n \times (25 - n)}\)

Now let's count the letters:
The problem states "a letter was sent in response to each report." This means there's exactly one letter for every report that was sent.

Since we just calculated there are \(\mathrm{n(25 - n)}\) reports, there must also be \(\mathrm{n(25 - n)}\) letters sent in response.
Number of letters = \(\mathrm{n \times (25 - n)}\)

Process Skill: APPLY CONSTRAINTS - Using the "one letter per report" constraint to determine the letter count

4. Combine and match to answer choices

Now we add up the total communications:

Total = Reports + Letters
Total = \(\mathrm{n(25 - n) + n(25 - n)}\)
Total = \(\mathrm{2 \times n(25 - n)}\)
Total = \(\mathrm{2n(25 - n)}\)

Looking at our answer choices:

1. \(\mathrm{n(25 - n)}\) - This is just the reports (or just the letters), not the total
2. \(\mathrm{n(26 - n)}\) - This has the wrong multiplier inside the parentheses
3. \(\mathrm{(n + 1)(25 - n)}\) - This changes the coefficient of \(\mathrm{(25 - n)}\)
4. \(\mathrm{n^2(25 - n)^2}\) - This squares both terms, which doesn't match our logic
5. \(\mathrm{2n(25 - n)}\) - This exactly matches our calculation!

Final Answer

The combined number of reports and letters sent is \(\mathrm{2n(25 - n)}\).

The answer is E.

Common Faltering Points

Errors while devising the approach

1. Misunderstanding which groups send communications

Students often confuse the direction of communication flow. They might think that non-task force members send reports TO task force members, when actually task force members send reports TO non-task force members. This fundamental misunderstanding of "who sends what to whom" leads to incorrect setup from the start.

2. Missing the two-way communication requirement

Many students focus only on the reports being sent and forget that the problem explicitly states "a letter was sent in response to each report." They might think they only need to count reports, missing that both reports AND letters need to be included in the total count.

3. Incorrectly calculating group sizes

Students may struggle with the complementary relationship between task force and non-task force members. Some might think there are 25 non-task force members instead of (25-n), forgetting that the task force members are part of the original 25 legislators.

Errors while executing the approach

1. Arithmetic errors while distributing multiplication

When calculating the total as \(\mathrm{n(25-n) + n(25-n)}\), students may make errors in combining like terms or factoring out common factors. They might incorrectly get \(\mathrm{n(50-2n)}\) instead of \(\mathrm{2n(25-n)}\), or make other algebraic manipulation mistakes.

2. Incorrectly applying the "one letter per report" constraint

Students might think that each task force member sends one letter total, rather than understanding that there's exactly one letter sent in response to each individual report. This leads to undercounting the number of letters.

Errors while selecting the answer

1. Selecting answer choice A by counting only one direction

Students who correctly calculate \(\mathrm{n(25-n)}\) for either reports OR letters but forget to add both components together will select choice A, which represents only half of the required total.

2. Confusing coefficient placement in the final expression

Even when students know they need to double their result, they might incorrectly place the factor of 2, potentially selecting choice C: \(\mathrm{(n+1)(25-n)}\) if they mistakenly think 2n should become (n+1), or make other similar algebraic rearrangement errors.

Alternate Solutions

Smart Numbers Approach

Step 1: Choose a specific value for n

Let's use \(\mathrm{n = 5}\) (task force has 5 members). This means there are \(\mathrm{25 - 5 = 20}\) committee members who are NOT on the task force.

Step 2: Count the reports sent

Each of the 5 task force members sends a report to each of the 20 non-task force members.
Number of reports = \(\mathrm{5 \times 20 = 100}\) reports

Step 3: Count the letters sent in response

For each report received, a letter is sent back. Since 100 reports were sent, 100 letters are sent in response.
Number of letters = 100 letters

Step 4: Calculate total communications

Total = Reports + Letters = \(\mathrm{100 + 100 = 200}\)

Step 5: Verify with answer choices using n = 5

Let's check which answer choice gives us 200 when \(\mathrm{n = 5}\):

• A. \(\mathrm{n(25 - n) = 5(25 - 5) = 5 \times 20 = 100}\) ❌
• B. \(\mathrm{n(26 - n) = 5(26 - 5) = 5 \times 21 = 105}\) ❌
• C. \(\mathrm{(n + 1)(25 - n) = (5 + 1)(25 - 5) = 6 \times 20 = 120}\) ❌
• D. \(\mathrm{n^2(25 - n)^2 = 5^2 \times 20^2 = 25 \times 400 = 10,000}\) ❌
• E. \(\mathrm{2n(25 - n) = 2 \times 5 \times 20 = 200}\) ✓

Step 6: Verify with a second value

Let's try \(\mathrm{n = 3}\) to confirm our answer:

• Task force: 3 members, Non-task force: 22 members
• Reports sent: \(\mathrm{3 \times 22 = 66}\)
• Letters sent: 66 (one per report)
• Total: \(\mathrm{66 + 66 = 132}\)

Checking answer choice E: \(\mathrm{2n(25 - n) = 2 \times 3 \times 22 = 132}\) ✓

The smart numbers approach confirms that answer choice E is correct.