Some context:
Our teacher gave us a classroom exercise and it was returned to us this week. I am revisiting all the questions in the exercise, particularly the ones I got wrong. I am stuck at a particular problem whose answer involves the derangement function.
Definition:
Derangement is defined using this function:
D(n, k) = the number of permutations where in n - k objects are not in their original position/s and k objects are in their original position/s.
Problem:
Jasmin, being a teacher’s pet, was asked to distribute the group assignments in the class. She plans to sabotage her classmates by exchanging the other group problems given to them. There are five groups in class including hers. In how many ways can she do this if she distributed the wrong problems to her classmates with her group receiving the correct problem?
Correct Answer (based on an answer key): D(5, 1)
My Question: Since Jasmin already has a group, aren't we supposed to only count the number of permutations in which the other 4 assignments will be given to the wrong group? If I'm not mistaken, D(5, 1) would find the sum of a, b, c, d, and e.
where:
a = the number of ways where only the other 4 assignments are given to the wrong group given that Jasmin is in group #1
b = the number of ways where only the other 4 assignments are given to the wrong group given that Jasmin is in group #2
c = the number of ways where only the other 4 assignments are given to the wrong group given that Jasmin is in group #3
d = the number of ways where only the other 4 assignments are given to the wrong group given that Jasmin is in group #4
e = the number of ways where only the other 4 assignments are given to the wrong group given that Jasmin is in group #5
All of those five would count the permutations where only 1 assignment is given to its correct group (i.e. Jasmin's group). But since the problem assumes that Jasmin already has a group, and is asking it in Jasmin's point of view, shouldn't the answer be D(4, 0)?
