Linear Arrangements
CAT’26 DILR CHEATSHEET
Every setup, rule and counting trick you need for CAT 2026 linear sets.
Linear arrangement is the friendliest looking part of CAT DILR and the one that quietly eats time once the clues start fighting each other. You are seating a handful of people in a single row, and the whole game is turning sentences like ‘two people sit between them’ or ‘she is somewhere to his left’ into exact seat numbers. Most of the pain is avoidable. It comes from misreading a facing direction, gluing people who were only loosely ordered, or branching on the wrong clue and drowning in cases. This sheet lays out the setups and rules you actually use, from fixed seats and gaps to blocks, counting, parallel rows and clean case splitting. Every box carries one worked mini puzzle, so you drill the move rather than just nodding at the idea.
Every linear arrangement rule you need
1Linear Setup and Total Ways
n distinct people fill n seats in a row, one each.
Total ways of n distinct items = n!
Example. 6 friends on chairs 1 to 6, with no clues yet, can sit in 6! = 720 ways.
CAT Hack. Number the seats 1 to n from left to right first, then drop every clue onto that fixed scale.
2Facing Direction and Left/Right
Facing north keeps left and right, facing south flips them.
North faces up, so left stays left. South faces down, so left and right swap
Example. In a north-facing row, A's right neighbour has a higher seat number. Turn the row to face south and A's right neighbour now has a lower number.
Common Mistake. Most left and right errors come from assuming everyone faces you. Read the facing line first.
3Fixed Position Clues
A clue that pins one person to a seat anchors the whole line.
X at seat k splits the row into left of k and right of k
Example. P sits at seat 4 in a row of 7, so three seats lie to P's left and three to the right, putting P exactly in the middle.
CAT Hack. Start from the most fixed clue, never a vague one. A pinned seat cuts the number of cases fastest.
4Immediate Neighbour Block
‘X is immediately next to Y’ glues them into one block.
X next to Y = one block [XY] or [YX]
Example. A is immediately next to B, so treat AB as one tile. In seats 1 to 5 it can start at 1, 2, 3 or 4, giving 4 placements, each with 2 internal orders.
CAT Favourite. Immediate-neighbour blocks appear in almost every CAT linear set, so spot and glue them at once.
5Gap Clues (K Between)
‘K people between X and Y’ fixes the seat gap between them.
Exactly K between X and Y → |seatX − seatY| = K + 1
Example. Two people between M and N means |seatM − seatN| = 3, so in seats 1 to 6 the pairs are (1,4), (2,5), (3,6) and their reverses.
CAT Insight. ‘Between’ counts only the people in the gap, so the seat difference is always one more than that.
6Ends and Extremes
A person at an end of the row has exactly one neighbour.
Ends are seats 1 and n, each with a single neighbour
Example. In a row of 8, R sits at an end and S is next to R, so S is fixed at seat 2 or seat 7, giving 2 options.
CAT Hack. Clues like ‘only one neighbour’ or ‘nobody to the left’ quietly place someone at an end.
7Ordering Clues (Somewhere Left)
‘X is somewhere left of Y’ sets order, not distance.
X left of Y → seatX < seatY, any gap allowed
Example. A is left of B and B is left of C, so the order A, B, C holds, yet in seats 1 to 5 they can still land at seats 1, 3 and 5.
Common Mistake. ‘Left of’ does not mean ‘immediately left of’, so do not glue them together.
8Group Together as a Block
k people who must sit together act as one super-seat.
k together among n = (n − k + 1)! × k!
Example. 3 of 6 people must sit together, so (6 − 3 + 1)! × 3! = 4! × 6 = 144 ways.
CAT Favourite. This block trick turns a messy ‘together’ condition into a clean two-step count.
9Not-Together Count
Count the lines where two named people never sit together.
Not together = n! − (n − 1)! × 2
Example. 5 people with A and B apart give 5! − (4! × 2) = 120 − 48 = 72 ways.
CAT Hack. Counting the forbidden ‘together’ case and subtracting beats listing every allowed line.
10Two Parallel Rows
Two rows face each other, so every seat has an opposite.
Row 1 faces row 2, seat i sits opposite seat i
Example. 6 people in two rows of 3 facing each other means row-1 seat 2 looks straight at row-2 seat 2, so their left and right are mirror images.
CAT Insight. The rows face each other, so one row’s left is the other row’s right. Mark facing before left and right clues.
11Case-Splitting on Ambiguity
When a clue allows two spots, branch into separate cases.
Clue with 2 options → build both cases, then test each
Example. X is at seat 2 or seat 6, so draw both lines, apply the rest of the clues, and drop the case that breaks a rule, often leaving one valid line.
CAT Hack. Split on the clue with the fewest options, usually two, so you carry the least branching.
12Position Elimination Grid
A person-by-seat grid tracks who can sit where.
Grid of people × seats, mark the impossible cells
Example. Cross out P at seat 1 once a clue bars it, and when a person’s row has one open seat left, that seat is forced.
CAT Favourite. The grid is the workhorse of hard linear sets, so build it before you start guessing positions.
13Max and Min Position
‘At least k to the left of X’ bounds where X can sit.
At least k people left of X → seatX ≥ k + 1
Example. At least 3 people sit to the left of Y in a row of 7, so Y sits at seat 4 or later, making Y’s earliest seat 4.
CAT Insight. One-sided clues rarely fix a seat outright, but they trim the range hard.
Exam shortcuts, common traps and revision
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CAT Exam Shortcuts
- Number the seats 1 to n from left to right before reading clues
- Read the facing line first, because it decides left and right
- Exactly K between two people means a seat gap of K + 1
- Glue immediate neighbours into a single block
- k people together = (n − k + 1)! × k!
- Not together = total − together
- Anchor the most fixed clue, then split on two-option clues
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Most Common CAT Traps
- Reading ‘left of’ as ‘immediately left of’ and gluing people wrongly
- Forgetting that a south-facing row flips left and right
- Counting ‘between’ as the seat gap instead of the gap minus one
- Missing that an end seat has only one neighbour
- Branching on the clue with the most options and drowning in cases
- Treating two facing rows as one long single line
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30-Second Revision Box
- Seats numbered 1 to n, one person in each
- North keeps left and right, south flips them
- K between means a seat difference of K + 1
- Immediate neighbour is a block, plain ‘left of’ is only order
- k together = (n − k + 1)! × k!, not together = total − together
- Anchor the fixed clue, then split cases on the smallest branch
The fastest way to get good at linear sets is to stop reading them and start drilling them, because the moves only turn automatic once your hand has done them a few dozen times. Anchor the fixed clue, respect the facing line, and split on the smallest branch, and most sets fall out in a couple of minutes. When you want to see where you actually stand, run a full mock and check your projected percentile with the Optima Learn CAT score predictor, then work through the DILR breakdowns on the blog. If you would rather build a plan with a person, book a free call with an Optima Learn mentor and shape one around your target.