In English Class, after studying a book called \The Lord of the Flies" by William Golding, students are split into 6 groups and each group is required to make a fi�lm based on one of the 8 important scenes in the novel. Each group can choose which scene they are creating a fi�lm of. However, to make sure there is a variety of scenes being created, there can only be a maximum of 2 groups creating the same scene. Find the last 3 digits of the number of possible ways there are for each group to choose a scene. pls give a detailed solution Details and Assumptions Each group has di�erent students in it, so therefore each group is consid- ered distinct

Question

In English Class, after studying a book called \The Lord of the Flies" by William Golding, students are split into 6 groups and each group is required to make a fi�lm based on one of the 8 important scenes in the novel. Each group can choose which scene they are creating a fi�lm of. However, to make sure there is a variety of scenes being created, there can only be a maximum of 2 groups creating the same scene. Find the last 3 digits of the number of possible ways there are for each group to choose a scene. pls give a detailed solution Details and Assumptions Each group has di�erent students in it, so therefore each group is consid- ered distinct

Answer 1

To find the number of possible ways for each group to choose a scene, we can use combinations.

Since there are 6 groups and each group can choose one of the 8 scenes, we need to find the combination of 8 scenes taken 6 at a time without repetition. This can be calculated using the formula for combinations:

C(n, r) = n! / (r!(n - r)!)

Where n is the total number of scenes (8) and r is the number of groups (6).

Using this formula, we can calculate the combination of scenes:

C(8, 6) = 8! / (6!(8 - 6)!) = 8! / (6!2!) = (8 * 7 * 6!) / (6! * 2 * 1) = (8 * 7) / (2 * 1) = 56 / 2 = 28

So there are 28 possible ways for each group to choose a scene.

However, we need to consider the constraint that there can only be a maximum of 2 groups creating the same scene. To account for this constraint, we can use combinations with repetition.

For the first group, there are 8 scenes to choose from. For the second group, since it cannot choose the same scene as the first group, there are 7 scenes left to choose from. The remaining groups can choose from the remaining scenes in a similar manner.

Using combinations with repetition, we can calculate the number of ways each group can choose a scene:

7 * 6 * 5 * 4 * 3 * 2 = 5040

So there are 5040 possible ways for each group to choose a scene, considering the constraint.

Finally, we need to find the last 3 digits of the total number of possible ways. To do this, we need to calculate the remainder when dividing the total number of ways (28) by 1000.

28 % 1000 = 28

Therefore, the last 3 digits of the total number of possible ways for each group to choose a scene is 028.