Arrangements with restrictions
- A restriction means some positions are fussy. Deal with the fussy part first.
- Together means glue the objects into one block.
- Apart / not next to usually means arrange the other objects first, then use the gaps.
Glue, gaps, or subtract
| Words | Move | What you count |
|---|---|---|
| together / consecutive | glue | treat the group as one block, then multiply by the order inside |
| not together | subtract | all arrangements minus the glued arrangements |
| no two adjacent | gaps | place the other objects first, then choose gaps |
| fixed end / fixed position | place first | lock that item, then arrange the rest |
| odd / even / greater than | split cases | handle the restricted digits first |
Worked example
First rung: two people together
A, B, C and D stand in a row. A and B must be together.
- Glue A and B into one block: . That gives 3 objects to arrange.
- Arrange the 3 objects in ways.
- Inside the block, A and B can be AB or BA, so multiply by.
Answer
Worked example
One extra step: not together
A, B, C and D stand in a row. A and B must not be together.
- Count all arrangements: .
- Subtract the together case from the previous example:.
Answer
Worked example
No two consonants together
In ANDROMEDA, the consonants D, D, M, N, R must not sit next to each other. (9709/52/O/N/23 Q7(a))
- Arrange the non-consonants A, A, E, O first:.
- Four non-consonants make 5 gaps around them. There are exactly 5 consonants, so each gap must get one.
- Arrange the consonants D, D, M, N, R in those gaps:.
- Multiply: .
Answer
Worked example
A at each end, Ds not together
Still using ANDROMEDA: put an A at each end, and keep the two Ds apart. (9709/52/O/N/23 Q7(b))
- With A fixed at both ends, arrange the middle 7 letters N, D, R, O, M, E, D:.
- Bad case: glue DD into one block with N, R, O, M, E:.
- Good case: .
Answer
Common mistake
For a glued block, remember the order inside the block. If two named people sit together, the block can be AB or BA, so there is a.
When subtracting is cleaner
- For not together, direct counting can get messy.
- It is often faster to count all arrangements, then subtract the together case.
- Make sure both counts use the same base: same letters, same fixed positions, same repeated-letter divisions.