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.
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. .