Question:
Five lectures L1, L2, L3, L4, L5 must be scheduled from Monday to Friday (one each day).
Five professors A, B, C, D, E will take one lecture each.
Constraints:
1. A takes L3.
2. L2 must be scheduled after L5.
3. C does not teach on Wednesday or Friday.
4. D teaches before E.
5. B does not teach L4.
How many valid schedules are possible?
Five lectures L1, L2, L3, L4, L5 must be scheduled from Monday to Friday (one each day).
Five professors A, B, C, D, E will take one lecture each. Constraints:
1. A takes L3.
2. L2 must be scheduled after L5.
3. C does not teach on Wednesday or Friday.
4. D teaches before E.
5. B does not teach L4.
How many valid schedules are possible?
Five professors A, B, C, D, E will take one lecture each. Constraints:
1. A takes L3.
2. L2 must be scheduled after L5.
3. C does not teach on Wednesday or Friday.
4. D teaches before E.
5. B does not teach L4.
How many valid schedules are possible?
Show Hint
Break scheduling problems into two layers:
1. Positioning events with ordering constraints
2. Assigning people under availability rules.
Count possibilities in each layer consistently.
1. Positioning events with ordering constraints
2. Assigning people under availability rules.
Count possibilities in each layer consistently.
Updated On: Nov 30, 2025
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The Correct Option is C
Solution and Explanation
We schedule lectures L1--L5 on Mon--Fri, and assign professors A--E (each exactly once).
Step 1: A must teach L3.
So L3 is fixed on whichever day A is assigned.
Step 2: Order restriction L2 after L5.
Valid (L5, L2) day-pairs:
(Mon,Tue), (Mon,Wed), (Mon,Thu), (Mon,Fri),
(Tue,Wed), (Tue,Thu), (Tue,Fri),
(Wed,Thu), (Wed,Fri),
(Thu,Fri).
A total of 10 possible placements.
Step 3: Professor restrictions.
C cannot teach on Wed or Fri -- C must be Mon/Tue/Thu.
B cannot teach L4 -- whichever day L4 lands on cannot get B.
D must teach before E -- day(D) < day(E).
Step 4: Combine lecture placements with professor placements.
For each of the 10 valid positions of (L5, L2), we must assign:
- L3 to A.
- Remaining lectures L1, L4, L5, L2 to professors B, C, D, E respecting rules.
In each arrangement:
• C may be assigned only if the lecture day is Mon/Tue/Thu.
• B cannot take L4.
• D and E must satisfy the day order.
Careful counting case-by-case (over each of the 10 allowed (L5, L2) placements) gives: \[ \text{Total valid schedules} = 18. \] Final Answer: 18
Step 1: A must teach L3.
So L3 is fixed on whichever day A is assigned.
Step 2: Order restriction L2 after L5.
Valid (L5, L2) day-pairs:
(Mon,Tue), (Mon,Wed), (Mon,Thu), (Mon,Fri),
(Tue,Wed), (Tue,Thu), (Tue,Fri),
(Wed,Thu), (Wed,Fri),
(Thu,Fri).
A total of 10 possible placements.
Step 3: Professor restrictions.
C cannot teach on Wed or Fri -- C must be Mon/Tue/Thu.
B cannot teach L4 -- whichever day L4 lands on cannot get B.
D must teach before E -- day(D) < day(E).
Step 4: Combine lecture placements with professor placements.
For each of the 10 valid positions of (L5, L2), we must assign:
- L3 to A.
- Remaining lectures L1, L4, L5, L2 to professors B, C, D, E respecting rules.
In each arrangement:
• C may be assigned only if the lecture day is Mon/Tue/Thu.
• B cannot take L4.
• D and E must satisfy the day order.
Careful counting case-by-case (over each of the 10 allowed (L5, L2) placements) gives: \[ \text{Total valid schedules} = 18. \] Final Answer: 18
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