To address the problem properly, let's break it down step by step:

Given Conditions:

1. 11 TAs in total:

   • 9 full-time TAs.
   • 2 half-time TAs who won't lead any assignment.

2. Assignments:

   • 7 assignments to be distributed among the TAs.
   • Each assignment must have:
     • 1 lead TA.
     • 2 assistant TAs.

3. Restrictions:

   • Each TA can be involved in at most 2 assignments.
   • Half-time TAs won't lead any assignment.
   • Two full-time TAs that didn't lead any of the 7 assignments will lead the midterm and final exam.

Steps to Solve:

1. Lead Assignment:

   • Since only full-time TAs can lead assignments, we need to assign 1 lead TA for each of the 7 assignments.
   • This leaves 2 full-time TAs who do not lead any of the 7 assignments (these two will lead the midterm and final).

2. Assistant Assignment:

   • We need to assign 2 assistant TAs per assignment.
   • Since there are 7 assignments, we need to assign 14 assistant spots.
   • Half-time TAs can assist but not lead, and since each TA can participate in a maximum of 2 assignments, we need to distribute the assistants carefully across the 11 TAs.

Solution Strategy:

1. Assigning Lead TAs:

   • Choose 7 full-time TAs to lead the assignments. This means that 2 full-time TAs will remain available for leading the midterm and final exam.

2. Assigning Assistant TAs:

   • We need 14 assistant roles across the assignments. These roles will be filled by a mix of the remaining 9 full-time and 2 half-time TAs.
   • Since each TA can only be involved in 2 assignments, we must ensure the assistant roles are assigned without exceeding this limit.

Breakdown of Assistants:

• We have 11 TAs in total, and each can participate in up to 2 assignments.
• The 7 full-time TAs who are leads can each take on 1 assistant role.
• The 2 remaining full-time TAs, who are not leading any assignments, can take 2 assistant roles each.
• The 2 half-time TAs can take 2 assistant roles each (but cannot lead).

This way, we will have:

• 7 assistant roles filled by the 7 full-time TAs who lead assignments.
• 4 assistant roles filled by the 2 full-time TAs who do not lead any assignment.
• 4 assistant roles filled by the 2 half-time TAs (2 each).

This satisfies the condition that each TA can participate in at most 2 assignments, and each assignment has exactly 1 lead TA and 2 assistants.

Final Roles:

• Leads: 7 full-time TAs (one lead per assignment).
• Assistants: 14 assistant roles distributed as described above.

Do you need further clarification on any part of this solution, or should I go into more detail?

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Further Questions:

1. How would the scheduling change if there were 8 assignments instead of 7?
2. What if each TA could participate in up to 3 assignments instead of 2?
3. What if the half-time TAs could lead one assignment but no more?
4. How would the assistant roles be redistributed if there were only 1 half-time TA?
5. What if only one full-time TA could be excluded from leading assignments?

Tip: When distributing resources or assignments with constraints, visualize or tabulate who takes on which role to ensure no one exceeds the limits.