2021 April MIMC 10 Problems/Problem 16
Find the number of permutations of 
such that at exactly two 
s are adjacent, and the 
s are not adjacent.
Solution
We can use casework counting to solve this problem.
The first case is 
. Since cannot be adjacent, then there are three such cases. there are 
for each of the case. However, cannot be adjacent, therefore, there are 
such arrangements.
The second case is 
. There are total of 
possible cases for B to not be adjacent. There are 
total possible such arrangements. By symmetrical counting, the first case is the same as 
and the second case is the same as 
.
The last we want to find is the number of arrangements of 
. For this case, there are total of possible placement of two s to avoid adjacency. Each has 
arrangement. Therefore, there are total of such arrangements.
.
