Let's break down the problem and solve it step-by-step. --- ## **Step 1: Understanding Time Slots** - **8M/8T:** 8:00–9:15 AM (Mon/Wed or Tue/Thu) - **9M/9T:** 9:30–10:45 AM - **11M/11T:** 11:00–12:15 PM - **12M/12T:** 12:30–1:45 PM - **14M/14T:** 2:00–3:15 PM Kathleen wants to take **six courses**: Math, History, English, Biology, Spanish, Psychology. --- ## **Step 2: Listing Preferences** Write the preference order for each course; the first is best (cost 1), the second is cost 2, etc. | Course | Available Sections (in order of preference) | |------------|-----------------------------------------------------| | Math | 11T (1), 12T (2), 9T (3), 11M (4), 12M (5), 9M (6), 8T (7), 8M (8) | | History | 11T (1), 11M (2), 14T (3), 14M (4), 8T (5), 8M (6) | | English | 9T (1), 11T (2), 14T (3), 11M (4), 12T (5), 14M (6), 12M (7), 9M (8)| | Biology | 14T (1), 11M (2), 12M (3), 14M (4), 9M (5), 8T (6), 8M (7)| | Spanish | 9T (1), 11M (2), 12M (3), 8T (4) | | Psychology | 14T (1), 11T (2), 12T (3), 9T (4), 14M (5), 8M (6) | --- ## **Step 3: Build the Schedule (Part a)** ### **Goal:** Assign one section per course, no time conflicts, minimizing total "cost". #### **Strategy:** - Assign the highest-preference available section for each course, making sure no two classes occupy the same slot. --- ### **Step 3A: Choosing the Best Schedule** Let’s try to greedily assign the highest-preference sections without conflicts: #### **1. Math: 11T** (cost 1) #### **2. History: 11M** (cost 2, since 11T is taken by Math) #### **3. English: 9T** (cost 1) #### **4. Biology: 14T** (cost 1) #### **5. Spanish: 12M** (cost 3, since 11M is taken by History) #### **6. Psychology: 12T** (cost 3, since 11T and 14T are taken) Let's check for time conflicts: - **Math: 11T (Tue/Thu 11:00–12:15)** - **History: 11M (Mon/Wed 11:00–12:15)** - **English: 9T (Tue/Thu 9:30–10:45)** - **Biology: 14T (Tue/Thu 2:00–3:15)** - **Spanish: 12M (Mon/Wed 12:30–1:45)** - **Psychology: 12T (Tue/Thu 12:30–1:45)** No conflicts! Let's sum the "cost": Math (1) + History (2) + English (1) + Biology (1) + Spanish (3) + Psychology (3) = **11** --- ## **Step 4: Lunch Break (Part b)** ### **Constraint:** No classes from 11:00–12:00. So, can't use **11M** or **11T**. Let's assign: #### **Math: 12T** (cost 2) #### **History: 14T** (cost 3) #### **English: 9T** (cost 1) #### **Biology: 14M** (cost 4) #### **Spanish: 12M** (cost 3) #### **Psychology: 12T** (cost 3) Check for conflicts: - Math: 12T (Tue/Thu 12:30–1:45) - History: 14T (Tue/Thu 2:00–3:15) - English: 9T (Tue/Thu 9:30–10:45) - Biology: 14M (Mon/Wed 2:00–3:15) - Spanish: 12M (Mon/Wed 12:30–1:45) - Psychology: 12T (Tue/Thu 12:30–1:45) **Conflict:** Math 12T and Psychology 12T are at the same time. So, let's try Psychology: 14M (cost 5, 2:00–3:15 Mon/Wed). Now, Biology and Psychology both want 14M. Try Biology: 12M (cost 3). Now, Spanish and Biology both want 12M. Try Spanish: 8T (cost 4). Final assignment: - Math: 12T (2) - History: 14T (3) - English: 9T (1) - Biology: 12M (3) - Spanish: 8T (4) - Psychology: 14M (5) Sum: 2 + 3 + 1 + 3 + 4 + 5 = **18** --- ## **Step 5: Two Days Only (Part c)** To have all classes on either M/W or T/Th, can have at most 5 courses per set, so must drop one. ### **Case 1: All on T/Th** List the highest-preference T/Th section for each course: - Math: 11T (1) - History: 11T (1) - English: 9T (1) - Biology: 14T (1) - Spanish: 9T (1) - Psychology: 14T (1) Now, let's check for time conflicts: - 9T: English, Spanish (conflict) - 11T: Math, History (conflict) - 14T: Biology, Psychology (conflict) So, can't take all six. Let's try removing one course at a time, and see which gives the lowest total cost. #### **Option: Drop Spanish** Math: 11T (1), History: 14T (3), English: 9T (1), Biology: 14T (1), Psychology: 12T (3) - 11T: Math - 14T: History, Biology (conflict) Try: Math: 11T (1), History: 8T (5), English: 9T (1), Biology: 14T (1), Psychology: 12T (3) - 11T: Math - 8T: History - 9T: English - 14T: Biology - 12T: Psychology No conflicts! Total cost: Math (1) + History (5) + English (1) + Biology (1) + Psychology (3) = **11** #### **Option: Drop English** Math: 11T (1), History: 14T (3), Biology: 14T (1), Spanish: 9T (1), Psychology: 12T (3) - 11T: Math - 14T: History, Biology (conflict) - 9T: Spanish - 12T: Psychology Try: History: 8T (5), Biology: 14T (1) - 8T: History - 14T: Biology No conflict. So, Math (1), History (5), Biology (1), Spanish (1), Psychology (3) = **11** #### **Option: Drop History** Math: 11T (1), English: 9T (1), Biology: 14T (1), Spanish: 9T (1), Psychology: 12T (3) But English and Spanish both want 9T. Try: English: 11T (2), Spanish: 9T (1) - 11T: Math, English (conflict) Try: English: 14T (3), Spanish: 9T (1) - 14T: Biology, English (conflict) Try: English: 12T (5), Spanish: 9T (1) - 12T: English, Psychology (conflict) Try: English: 12T (5), Psychology: 14T (1) - 12T: English - 14T: Biology, Psychology (conflict) Looks like the best option is to drop Spanish or English, both yield a cost of **11**. --- ### **Case 2: All on Mon/Wed** List M/W sections: - Math: 11M (4), 12M (5), 9M (6), 8M (8) - History: 11M (2), 14M (4), 8M (6) - English: 11M (4), 14M (6), 12M (7), 9M (8) - Biology: 11M (2), 12M (3), 14M (4), 9M (5), 8M (7) - Spanish: 11M (2), 12M (3) - Psychology: 14M (5), 8M (6) Try to assign highest preferences, check for conflicts: Let's try: - Math: 11M (4) - History: 14M (4) - English: 11M (4) - Biology: 12M (3) - Spanish: 12M (3) - Psychology: 14M (5) Check for conflicts: - 11M: Math, English (conflict) - 12M: Biology, Spanish (conflict) - 14M: History, Psychology (conflict) Need to drop a course. Let's try dropping Spanish. Now assign: - Math: 11M (4) - History: 14M (4) - English: 11M (4) - Biology: 12M (3) - Psychology: 14M (5) Still: 11M: Math, English (conflict), 14M: History, Psychology (conflict) Try: - Math: 9M (6) - History: 14M (4) - English: 11M (4) - Biology: 12M (3) - Psychology: 14M (5) So, - 9M: Math - 14M: History, Psychology (conflict) - 11M: English - 12M: Biology Try Psychology: 8M (6) - 8M: Psychology - 9M: Math - 11M: English - 12M: Biology - 14M: History No conflicts. Sum cost: Math (6) + History (4) + English (4) + Biology (3) + Psychology (6) = **23** --- ## **Summary Table** | Scenario | Courses Dropped | Total Cost | Notes | |-----------------------------|----------------|------------|-------------------------------| | (a) No constraint | None | 11 | All six, no conflicts | | (b) Lunch 11-12 open | None | 18 | All six, no conflicts | | (c) All on TTh (drop 1) | Spanish/English| 11 | Drop Spanish or English | | (c) All on MW (drop 1) | Spanish | 23 | Drop Spanish | --- ## **Final Answers** ### **(a) Most Preferred Schedule** - **Math:** 11T - **History:** 11M - **English:** 9T - **Biology:** 14T - **Spanish:** 12M - **Psychology:** 12T ### **(b) With 11:00-noon free** - **Math:** 12T - **History:** 14T - **English:** 9T - **Biology:** 12M - **Spanish:** 8T - **Psychology:** 14M ### **(c) All Classes on Two Days** - **T/Th (drop Spanish or English):** - Math: 11T - History: 8T - English: 9T (if keep English), or Psychology: 12T (if keep Spanish) - Biology: 14T - Psychology: 12T - (Total cost: 11) - **M/W (drop Spanish):** - Math: 9M - History: 14M - English: 11M - Biology: 12M - Psychology: 8M - (Total cost: 23) **T/Th schedule matches preferences much more closely than M/W.** --- If you want the actual weekly schedule table, let me know!

Let's go through each part of the problem step-by-step, detailing the reasoning, calculations, and final answers. --- ## **Part (a): Determine a class schedule that most closely meets Kathleen's preferences** ### **Step 1: Understand the data** Kathleen wants to take 6 courses with the following available sections, ordered by her preference: | Course | Sections (Preference Order) | |------------|--------------------------------------------------| | Math | 11T(1), 12T(2), 9T(3), 11M(4), 12M(5), 9M(6), 8T(7), 8M(8) | | History | 11T(1), 11M(2), 14T(3), 14M(4), 8T(5), 8M(6) | | English | 9T(1), 11T(2), 14T(3), 11M(4), 12T(5), 14M(6), 12M(7), 9M(8) | | Biology | 14T(1), 11M(2), 12M(3), 14M(4), 9M(5), 8T(6), 8M(7) | | Spanish | 9T(1), 11M(2), 12M(3), 8T(4) | | Psychology | 14T(1), 11T(2), 12T(3), 9T(4), 14M(5), 8M(6) | ### **Step 2: Assign "cost" to each section based on preference** - 1st choice: cost 1 - 2nd choice: cost 2 - 3rd choice: cost 3 - and so on... ### **Step 3: Find the optimal schedule with no conflicts** Goal: Assign one section per course, no overlapping class times, minimizing total "cost" (sum of preference weights). --- ### **Step 4: Trial assignment** We'll try assigning the **most preferred sections** first and check for conflicts. 1. **Math:** 11T (cost 1) — **Monday/Wednesday 11:00–12:15** 2. **History:** 11M (cost 2) — **Monday/Wednesday 11:00–12:15** Conflict? Yes, both at 11:00–12:15 on different days, so no conflict. 3. **English:** 9T (cost 1) — **Tuesday/Thursday 9:30–10:45** No conflict with Math and History. 4. **Biology:** 14T (cost 1) — **Tuesday/Thursday 2:00–3:15** No conflict. 5. **Spanish:** 9T (cost 1) — **Tuesday/Thursday 9:30–10:45** Conflict with English (both at 9:30–10:45). To resolve, pick English: 11T (cost 2). Now, Spanish: 12M (cost 3) — **Monday/Wed 12:30–1:45** 6. **Psychology:** 14T (cost 1) — **Tuesday/Thursday 2:00–3:15** Conflict with Biology (both at 2:00–3:15). Choose Psychology: 14M (cost 5). --- ### **Step 5: Final assignment for part (a)** | Course | Assigned Section | Time Slot | Preference Cost | |------------|--------------------|----------------------------------|-----------------| | Math | 11T | Tue/Thu 11:00–12:15 | 1 | | History | 11M | Mon/Wed 11:00–12:15 | 2 | | English | 11T (second choice)| Tue/Thu 11:00–12:15 | 2 (conflict with History, so pick different) | | Biology | 14T | Tue/Thu 2:00–3:15 | 1 | | Spanish | 12M | Mon/Wed 12:30–1:45 | 3 | | Psychology | 14M | Mon/Wed 2:00–3:15 | 5 | But **conflict exists**: English 11T and Math 11T occur at the same time (Tue/Thu 11:00–12:15). To fix this, select English's **next preference**. **English options:** - 14T (cost 3): conflict with Biology (also at 2:00–3:15), conflicts with Biology. - 12T (cost 5): time conflict with Math 11T (already assigned). - 11M (cost 4): conflict with Math 11T. - 12T (cost 5): conflict with Math 11T. - 9T (cost 1): **No conflict**: same time as Math 11T? Let's check: - 9T: Tuesday/Thursday 9:30–10:45 — **no conflict**. So, assign English: 9T (cost 1). Similarly, for Spanish, choose the highest preference that doesn't conflict: - 12M (cost 3): Monday/Wed 12:30–1:45, no conflict with others assigned. - Psychology: 14M (cost 5): Monday/Wed 2:00–3:15, conflicts with Biology 14T? Biology is 14T (Tue/Thu 2:00–3:15). Conflict! So, pick Psychology: 14T (cost 1). But we already assigned Biology: 14T, so conflict exists. Next preference for Psychology: - 12T (cost 3): conflicts with Math 11T? Both at 11:00–12:15, at different days, so no conflict. - 14M (cost 5): conflict with Biology 14T. - 8M (cost 8): 8M is 8:00–9:15, no conflict. Choose Psychology: 8M (cost 8). --- ### **Final schedule with minimized total preference cost:** | Course | Section | Time Slot | Preference Cost | |------------|---------|----------------------------------|-----------------| | Math | 11T | Tue/Thu 11:00–12:15 | 1 | | History | 11M | Mon/Wed 11:00–12:15 | 2 | | English | 9T | Tue/Thu 9:30–10:45 | 1 | | Biology | 14T | Tue/Thu 2:00–3:15 | 1 | | Spanish | 12M | Mon/Wed 12:30–1:45 | 3 | | Psychology | 8M | Mon/Wed 8:00–9:15 | 8 | **Total cost:** 1 + 2 + 1 + 1 + 3 + 8 = **16** **Note:** This is a reasonable schedule that respects her preferences and avoids conflicts. --- ## **Final answer for part (a):** **Schedule:** - Math: 11T (Tue/Thu 11:00–12:15) - History: 11M (Mon/Wed 11:00–12:15) - English: 9T (Tue/Thu 9:30–10:45) - Biology: 14T (Tue/Thu 2:00–3:15) - Spanish: 12M (Mon/Wed 12:30–1:45) - Psychology: 8M (Mon/Wed 8:00–9:15) --- ## **Part (b): If Kathleen wants to keep 11:00 a.m. to noon open for lunch** ### **Step 1: Constraint** Classes cannot be scheduled between 11:00–12:00. ### **Step 2: Adjust preferences** Eliminate all sections that occur during 11:00–12:00: - **Math:** 11T (11:00–12:15) → conflicts with lunch hour → **cannot choose 11T** - **History:** 11M (11:00–12:15) → conflict → **cannot choose 11M** - **English:** 11T (11:00–12:15) → conflict → **cannot choose 11T** - **Biology:** 11M (11:00–12:15) → conflict → **cannot choose 11M** - **Spanish:** 11M (11:00–12:15) → conflict → **cannot choose 11M** - **Psychology:** 11T (11:00–12:15) → conflict → **cannot choose 11T** Remaining options: - Math: 12T(2), 9T(3), 8T(7) - History: 14T(3), 14M(4), 8T(6) - English: 14T(3), 12T(5), 14M(6), 12M(7), 9T(1) - Biology: 14T(1), 12M(3), 14M(4), 9M(5), 8T(6) - Spanish: 12M(3), 8T(4) - Psychology: 14T(1), 12T(3), 14M(5), 8M(6) ### **Step 3: Assign sections avoiding 11:00–12:00** Start with highest preferences: - Math: 12T (cost 2) - History: 14T (cost 3) - English: 9T (cost 1) - Biology: 14T? Conflict with History (both at 14T). Let's pick Biology: 12M (cost 3) - Spanish: 8T (cost 4) - Psychology: 14T? Conflict with History. Next option: 12T (cost 3) Check for conflicts: - Math: 12T (Tue/Thu 12:30–1:45) - History: 14T (Tue/Thu 2:00–3:15) - English: 9T (Tue/Thu 9:30–10:45) - Biology: 12M (Mon/Wed 12:30–1:45) - Spanish: 8T (Tue/Thu 8:00–9:15) - Psychology: 12T (Tue/Thu 12:30–1:45) Conflict? Both Math and Psychology at 12:30–1:45 on Tuesday/Thursday. Conflict. Try Psychology: 14M (cost 5): - 14M: Mon/Wed 2:00–3:15 (no conflict with others). Now, review conflicts: - Math: 12T - Psychology: 14M - No conflicts. Sum cost: 2 + 3 + 1 + 3 + 4 + 5 = **18** --- ## **Final schedule for part (b):** | Course | Section | Time Slot | Preference Cost | |------------|---------|----------------------------------|-----------------| | Math | 12T | Tue/Thu 12:30–1:45 | 2 | | History | 14T | Tue/Thu 2:00–3:15 | 3 | | English | 9T | Tue/Thu 9:30–10:45 | 1 | | Biology | 12M | Mon/Wed 12:30–1:45 | 3 | | Spanish | 8T | Tue/Thu 8:00–9:15 | 4 | | Psychology | 14M | Mon/Wed 2:00–3:15 | 5 | **Total preference cost:** **18** --- ## **Part (c): Schedule on two days only (either M/W or T/Th)** ### **Step 1: Constraints** - She can only have classes on two days. - Only 5 time slots per day, so only 10 classes total. - She wants **all 6 courses**, so she must drop 1 course. ### **Step 2: Find the best schedule for T/Th** List her top preferences for T/Th: | Course | Preferred Sections (in order) | Time Slot | |------------|------------------------------|----------------------------------| | Math | 11T(1), 12T(2), 9T(3), 14T(4), 8T(5) | Tue/Thu times | | History | 11T(1), 14T(3), 8T(5) | Tue/Thu times | | English | 9T(1), 11T(2), 14T(3), 12T(5), 8T(6) | Tue/Thu times | | Biology | 14T(1), 11T(2), 12T(3), 14M(5), 8M(6) | Tue/Thu times | | Spanish | 12M(1), 11M(2), 8T(4) | Mon/Wed times, so ignore for T/Th | | Psychology | 14T(1), 11T(2), 12T(3), 9T(4), 14M(5) | Tue/Thu times | She can select: - Math: 11T (cost 1) - History: 14T (cost