Permutations with restrictions
Master permutations with restrictions for Year 11 Specialist Mathematics in Queensland (QCAA). They are ordered arrangements of distinct objects where a rule, such as keeping certain objects together or apart, limits which arrangements count.
You will learn the block method for objects that must stay together, the gap method for keeping objects apart, and how to count alternating and fixed-position arrangements — key skills for probability and QCAA combinatorics.
Theory
Permutations with restrictions are ordered arrangements of distinct objects where a condition limits which arrangements are allowed — a core skill of Year 11 Specialist Mathematics (QCAA, Queensland). This page shows the block method for keeping objects together, the gap method for keeping them apart, and how to handle alternating and fixed-position problems.
A permutation is an ordered arrangement of distinct objects. With no conditions, \(n\) distinct objects arrange in a row in \(n!\) ways. A restriction is a rule — such as "these two must be together" or "no two of these may touch" — that rules some of those arrangements out.
When certain objects must stay together, use the block method: tie those objects into a single unit (a "block"), arrange the units, then arrange the objects inside the block. Treating \(k\) objects as one block turns \(n\) objects into \(n-k+1\) units.
When objects must be kept apart (not next to each other), you have two routes. Subtract: count every arrangement, then take away the ones where they are together. Or use the gap method: seat the other objects first, then drop the restricted objects into the separate gaps between and around them, so no two can touch.
Two more common restrictions: alternating arrangements (two kinds of object must take turns), counted by fixing the pattern then arranging each kind; and fixed positions (an object is pinned to a particular seat or to an end), counted by locking that object in place and arranging the rest.
Objects together (block method). Tie the \(k\) objects that must stay together into one block. The \(n\) objects become \(n-k+1\) units; arrange the units, then the block internally:
Objects apart (subtraction). Take the together-count away from the total:
Objects apart (gap method). Seat the \(m\) other objects (\(m!\) ways). They create \(m+1\) gaps; slot the \(k\) restricted objects into separate gaps, order mattering:
Alternating (two kinds, \(p\) of one and \(q\) of the other). Count the valid patterns (\(2\) when \(p=q\), otherwise \(1\)), then arrange each kind:
How to count a restricted arrangement
- Name the restriction: are objects forced together, kept apart, made to alternate, or fixed to a position or end?
- Pick the technique: block method for together; subtraction or the gap method for apart; pattern-count for alternating; lock-then-arrange for a fixed position.
- Count each stage: the units (or seated objects and their gaps), and any internal arrangement of a block — watch that a block still needs \(\times k!\).
- Combine: multiply the stages together; for "apart" by subtraction, take the together-count away from the total.
Tie Jack and Kim into one block, then arrange the units:
| \(\text{units (block + 3 others)}\) | \(=\) | \(4\) |
| \(\text{arrange the units}\) | \(=\) | \(4!\) |
| \(\text{Jack and Kim inside the block}\) | \(=\) | \(2!\) |
| \(\text{total}\) | \(=\) | \(4! \times 2!\) |
| \(=\) | \(24 \times 2\) | |
| \(=\) | \(48\) |
\(48\) arrangements.
Count every arrangement, then subtract the together ones from Example 1:
| \(\text{all arrangements}\) | \(=\) | \(5!\) |
| \(=\) | \(120\) | |
| \(\text{together (block)}\) | \(=\) | \(4! \times 2!\) |
| \(=\) | \(48\) | |
| \(\text{apart}\) | \(=\) | \(120 - 48\) |
| \(=\) | \(72\) |
\(72\) arrangements.
Seat the boys first, then drop the girls into the separate gaps between them:
| \(\text{seat the 4 boys}\) | \(=\) | \(4!\) |
| \(=\) | \(24\) | |
| \(\text{gaps created (before, between, after)}\) | \(=\) | \(5\) |
| \(\text{place 3 girls in the gaps (order matters)}\) | \(=\) | \(5 \times 4 \times 3\) |
| \(=\) | \(60\) | |
| \(\text{total}\) | \(=\) | \(24 \times 60\) |
| \(=\) | \(1440\) |
\(1440\) arrangements.
Count the patterns first, then arrange each gender within its own places:
| \(\text{patterns (BGBGBG, GBGBGB)}\) | \(=\) | \(2\) |
| \(\text{arrange the boys}\) | \(=\) | \(3!\) |
| \(\text{arrange the girls}\) | \(=\) | \(3!\) |
| \(\text{total}\) | \(=\) | \(2 \times 3! \times 3!\) |
| \(=\) | \(2 \times 6 \times 6\) | |
| \(=\) | \(72\) |
\(72\) arrangements.
Common pitfalls
Frequently asked questions
What is the block method for permutations?
When objects must stay together, tie them into a single block and treat it as one unit. Arrange the units, then arrange the objects inside the block. For \(k\) objects together among \(n\), that is \((n-k+1)!\times k!\).
How do you arrange objects so that two are not together?
Either subtract the together-count from the total number of arrangements, or use the gap method: seat the other objects first, then place the restricted objects into separate gaps so they cannot touch.
What is the gap method?
Seat the unrestricted objects first. Between and around them there are gaps — one more gap than the number seated. Drop the restricted objects into separate gaps, and because order matters this is a permutation of the gaps.
How many ways can boys and girls alternate in a row?
Count the valid alternating patterns first (two patterns when the numbers are equal, one otherwise), then multiply by the arrangements of each group. For three boys and three girls it is \(2\times3!\times3!=72\).
How do you handle a person fixed at one end of a row?
Lock that person into the fixed place, which happens in \(1\) way, then arrange everyone else. For six people with one fixed at the left end that is \(5!=120\) arrangements.
Why do you multiply by k! inside a block?
The block fixes which objects sit together, but not their order within the group. Those \(k\) objects can still be arranged among themselves in \(k!\) ways, so every unit-arrangement splits into \(k!\) real arrangements.