Question
If there are six period in each working day of a school , in how many ways can one arrange 5 subject such that each subject is allowed at least one period ?
Answers
GPT 3.5
To solve this problem, we can use the concept of stars and bars.
Let's say there are 5 subjects: A, B, C, D, and E. We want to arrange these subjects in 6 periods such that each subject has at least one period.
To ensure that each subject has at least one period, we can first distribute one period to each subject. So each subject now has 1 period assigned to it, and we are left with 1 additional period to distribute.
Now, we can use stars and bars to distribute the remaining period among the subjects. We can think of the additional period as a "star," and the areas between the subjects as "bars" that separate them. We need to distribute the star among the bars in order to assign the remaining period.
We have 5 subjects and 6 periods, so there are 5-1=4 bars. We need to distribute the star (additional period) among these 4 bars.
Using the stars and bars formula, the number of ways to distribute the star among the bars is given by (n+k-1) choose (k-1), where n is the number of stars and k is the number of bars.
In this case, we have 1 star and 4 bars, so the number of ways to distribute the periods is (1+4-1) choose (4-1) = 4 choose 3 = 4.
Therefore, there are 4 ways to arrange the 5 subjects such that each subject is allowed at least one period.
Let's say there are 5 subjects: A, B, C, D, and E. We want to arrange these subjects in 6 periods such that each subject has at least one period.
To ensure that each subject has at least one period, we can first distribute one period to each subject. So each subject now has 1 period assigned to it, and we are left with 1 additional period to distribute.
Now, we can use stars and bars to distribute the remaining period among the subjects. We can think of the additional period as a "star," and the areas between the subjects as "bars" that separate them. We need to distribute the star among the bars in order to assign the remaining period.
We have 5 subjects and 6 periods, so there are 5-1=4 bars. We need to distribute the star (additional period) among these 4 bars.
Using the stars and bars formula, the number of ways to distribute the star among the bars is given by (n+k-1) choose (k-1), where n is the number of stars and k is the number of bars.
In this case, we have 1 star and 4 bars, so the number of ways to distribute the periods is (1+4-1) choose (4-1) = 4 choose 3 = 4.
Therefore, there are 4 ways to arrange the 5 subjects such that each subject is allowed at least one period.