Arrangements with restrictions
A restriction changes which arrangements are legal. Translate the wording into a block, a gap, a fixed position or a complement before pressing any calculator button.
Change the objects or the spaces
A and B together
[AB or BA] [C] [D] [E]
4! × 2!
A and B not adjacent
gap C gap D gap E gap
put A and B in different gaps
Together means one block
- Glue the required objects into one block.
- Arrange the block with the other objects.
- Multiply by the possible orders inside the block.
Worked example
A and B together among five people
The objects are now : four objects in total.
Common mistake
Not together is often total minus together
“A and B are not together” is the complement of “A and B are together”. Both counts must use the same people and fixed positions.
Worked example
A and B apart among five people
- All arrangements: .
- Together arrangements: .
For several objects that must never be adjacent, arrange the other objects first and place the restricted objects in different gaps.
Worked example
Consonants separated in ANDROMEDA
Keep the five consonants D, D, M, N and R apart. (9709/52/O/N/23 Q7(a))
- Arrange A, A, E and O: .
- These four vowels make five gaps, including the two end gaps.
- Put one consonant in each gap. Their distinct orders number .
Turn an exact gap into position pairs
“Exactly three objects between A and B” means their position numbers differ by 4. List the legal position pairs before arranging anything else. This is different from treating A and B as one block.
Worked example
Exactly three people between A and B
Eight different people stand in a line.
- The position pairs are (1,5), (2,6), (3,7) and (4,8): 4 choices.
- A and B can swap positions: 2 choices.
- Arrange the other 6 people in the remaining positions.
Key idea
Seven different people stand in a line. A and B must have exactly two people between them. Find the number of arrangements.
Show worked answer
Their positions differ by 3. The legal pairs are (1,4), (2,5), (3,6) and (4,7).