Let's analyze the problem step-by-step:
Problem Restatement
- There are 9 professors.
- Each day, 3 professors give lectures.
- Every possible combination of 3 professors out of the 9 is used exactly once (no repetition).
- We want to find out how many days each professor will have to come.
Step 1: Calculate the total number of days
The total number of days corresponds to the total number of unique combinations of 3 professors chosen from 9. This is given by the combination formula:
(93)=9!3!(9−3)!=9×8×73×2×1=84\binom{9}{3}=\frac{9!}{3!(9-3)!}=\frac{9\times 8\times 7}{3\times 2\times 1}=84(39)=3!(9−3)!9!=3×2×19×8×7=84
So, there will be 84 days in total.
Step 2: Calculate how many days each professor must come
Each day, 3 professors come. Over 84 days, the total "professor slots" are:
84×3=25284\times 3=25284×3=252
Since there are 9 professors, and these slots are distributed evenly (because all combinations are used exactly once), the number of days each professor appears is:
2529=28\frac{252}{9}=289252=28
Final Answer:
Each professor will have to come for 28 days. If you'd like, I can also explain the reasoning behind why the distribution is even, or provide more insights!
