To solve this problem, we need to find the number of ways to arrange five subjects across six periods in a school day, with the condition that each subject must be assigned at least one period.

Step-by-step Solution:

1. Understanding the Problem:

   • We have 5 subjects: Let's denote them as $S_1, S_2, S_3, S_4, S_5$.
   • There are 6 periods in a day.
   • Each subject must be assigned to at least one period.

2. Approach:

   • We need to distribute 6 periods among 5 subjects such that no subject gets 0 periods.
   • This can be treated as a problem of distributing indistinguishable objects (periods) into distinguishable boxes (subjects), with each box getting at least one object.

3. Applying the Stars and Bars Theorem:

   • First, assign 1 period to each of the 5 subjects to satisfy the condition that each subject gets at least 1 period.
   • This consumes 5 periods, leaving us with $6 - 5 = 1$ period to distribute freely among the 5 subjects.

4. Finding the Number of Ways:

   • The remaining 1 period can be assigned to any of the 5 subjects. This is a simple selection problem.
   • Therefore, there are 5 ways to distribute the remaining period.

5. Arranging the Subjects within the Periods:

   • Once the periods are assigned to the subjects, we can arrange the subjects within the periods.
   • The total number of arrangements of 6 periods where each subject has at least one period is given by the formula: $\text{Total Ways} = 6! \times \text{(Number of ways to distribute periods)}$
   • Since there are 5 ways to distribute the periods (as calculated above), we have: $\text{Total Ways} = 6! \times 5$

6. Calculating the Result:

   • Calculate $6!$ (6 factorial): $6! = 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 720$
   • Now multiply by 5: $\text{Total Ways} = 720 \times 5 = 3600$

Conclusion:

There are 3,600 ways to arrange 5 subjects in 6 periods such that each subject is assigned at least one period.

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Would you like more details, or do you have any questions about this solution?

Related Questions:

1. How would the solution change if there were 7 periods in a day instead?
2. What if each subject could be assigned to a maximum of 2 periods?
3. How many ways are there to arrange 4 subjects in 5 periods such that each subject is assigned at least one period?
4. What if the periods are distinguishable and the order of subjects matters?
5. How many ways can 6 subjects be arranged in 6 periods with at least one period per subject?

Tip:

When dealing with combinatorial problems, always break them down into smaller, manageable parts and use combinatorial principles like "Stars and Bars" or permutations and combinations to solve them.