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.
.