Adjacency in Permutations
Lesson · Intermediate
Combinatorics
Adjacent objects → Block Method
Place a box around adjacent objects and treat them as a single object.
Non-adjacent objects → Gap Method
Place the other elements first and then place non-adjacent objects between them.
In how many permutations of the letters in LOGARITHM are adjacent?
There are 9 distinct letters.
Treat as one block:
We now have 7 objects:
These can be arranged in
ways.
The letters can be arranged inside the block in
ways.
Therefore
In how many permutations of
are no two of adjacent?
First arrange the other five objects:
There are
ways.
They create 6 gaps:
Choose 3 gaps:
Arrange :
Therefore
In how many permutations of LOGARITHM are adjacent and no two of are adjacent?
Consider as a box and ignore
There are
arrangements.
They create 6 gaps:
Choose 3 gaps:
Arrange :
Therefore
If must be pairwise non-adjacent, we cannot use
Here
counts cases where all three are together.
So its complement still includes cases where only two are adjacent, such as
Thus