if there is  six periods in each working day of school in howmany ways can  one arrange five subjects such that each sub. is allowed at least 1 period

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.

If there are six periods in each working day of school, in how many ways can one arrange five subjects such that each subject is allowed at least 1 period?

Explore how to arrange 5 subjects in 6 school periods such that each subject is assigned at least one period. This combinatorial problem uses the Stars and Bars theorem and permutation formulas. Suitable for students in Grades 10-12.

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