Arrangements & Grouping: Notes & Examples

BEGINNER

Arrangements & Grouping: visual worked lesson

Core idea

An arrangement places objects or people in positions. A grouping puts them into sets. Translate each clue into a clear restriction.

Solving method

Use fixed positions and linked blocks. In a circle, fix one person to remove equivalent rotations only when seats are otherwise unlabelled.

Worked example

A, B and C sit in a row. B is in the middle and A is left of B. Find the order. 1. Place B in the second seat. 2. Only the first seat is left of B, so put A there. 3. C takes the remaining seat: A, B, C.

Read the diagram

The boxes are seats viewed from left to right. Place B in the middle because that location is fixed. With only one seat to B’s left, A must occupy it. C then takes the remaining seat. In facing-direction questions, distinguish a person’s left from the viewer’s left.

Exam-style application

Five people A,B,C,D,E stand in a row. A is immediately before B; C is at an end; D is not beside E. Count arrangements. Treat AB as a block. With C fixed at either end there are 3!=6 arrangements of AB,D,E. Of these, D and E are adjacent in 2 x 2!=4 arrangements. Thus 2 valid arrangements per end, giving 4 total. Check one example: C-D-A-B-E.

Common mistakes

Keep track of whether left and right are from the viewer’s position or the seated person’s position. Block counting handles immediate adjacency, not merely "somewhere before". If seats have labels, rotational symmetry cannot simply be divided out.

Fill fixed seats before the remaining seat. From the viewer’s left: A – B – C. — Fill fixed seats before the remaining seat

From the viewer’s left: A – B – C. — Fill fixed seats before the remaining seat

Source record: Bharatiya Pathshala — original illustrated lessons.

BEGINNER

Core idea

Arrangement problems ask you to place people or objects in positions. Grouping problems ask which items belong together. Rules reduce the possible choices. A row can be drawn as empty boxes. A round table can be drawn as a circle with marked seats.

BEGINNER

Steps

1. Draw the positions. 2. Place anything whose position is fixed. 3. Join items that must stay together into a block. 4. Mark rules such as "not next to". 5. Try a small number of cases and reject a case as soon as it breaks a rule.

BEGINNER

Worked example

Ali, Bela, and Chet sit in three seats from left to right. Ali is at the left end, and Bela is beside Ali. Ali must be first, Bela second, and Chet third. All rules should still be true when you read the final row.

BEGINNER

Common mistake

For people facing each other or facing inward around a table, their left and right are not always your left and right on the page. Decide the direction before placing them.

Circular Arrangements

Fix one person to remove rotations. Treat reflections as different unless the question explicitly identifies them.

Worked example

Four distinct people around a round table have (4-1)! = 6 arrangements. Fix A, then arrange B, C and D in the three remaining positions.

Grouping

Grouping allocates objects to sets; arrangement orders them. First satisfy capacity and membership rules, then consider any order within groups.

Worked example

Choose a group of two from A, B, C and D, with A and B forbidden together. There are 6 pairs in total and one forbidden pair, leaving 5.

Application and strategy

Use fixed positions and linked blocks. In a circle, fix one person to remove equivalent rotations only when seats are otherwise unlabelled.

Worked application

Five people A,B,C,D,E stand in a row. A is immediately before B; C is at an end; D is not beside E. Count arrangements. Treat AB as a block. With C fixed at either end there are 3!=6 arrangements of AB,D,E. Of these, D and E are adjacent in 2 x 2!=4 arrangements. Thus 2 valid arrangements per end, giving 4 total. Check one example: C-D-A-B-E.

Common traps and strategy

Block counting handles immediate adjacency, not merely "somewhere before". If seats have labels, rotational symmetry cannot simply be divided out.

Grouping with membership and capacity rules

Core idea

Grouping assigns members to teams; arranging also determines order. Write each team's capacity and distinguish named teams from interchangeable groups before counting.

Solving method

Use one row per team. Record people who must be together as a block, and forbidden pairs as exclusions. Place fixed memberships first, then branch on the most restrictive remaining choice. Test every completed allocation against every rule.

Worked example

Select two people from A, B, C and D, with A and B forbidden together. The six possible pairs are AB, AC, AD, BC, BD and CD. Removing AB leaves five. If instead A may be selected only when B is selected, the valid pairs are AB, BC, BD and CD: the requirement does not force A whenever B is present.

Common mistakes

Do not count the order within an unordered team. A requirement in one direction is not automatically reversible. A team's capacity must still be satisfied after every placement.

Source record: Bharatiya Pathshala — reviewed preparation lessons.

Linear, circular and facing arrangements

Core idea

Draw numbered positions and mark each person's facing direction. Left and right belong to the person described. “Immediately left” means an adjacent position; “somewhere left” permits a gap.

Solving method

For a north-facing row, left is west on a north-up page; for a south-facing row, it is east. Two rows facing one another have opposite left-right orientations. In a circle, a person facing the centre has their left clockwise and right anticlockwise; these reverse when facing outward. Do not assume a facing direction unless the question supplies a convention.

Worked example

Four distinct people A, B, C and D sit in a north-facing row with A immediately left of B. Treat AB as one block. The three units AB, C and D have 3! = 6 orders. Around an unnumbered round table, fixing A removes rotational duplicates, leaving (4−1)! = 6 orders without other restrictions. Reflections remain different unless explicitly identified.

Common mistakes

For clockwise seats N, E, S, W facing inward, the person at N has E on their own left. A sketch's apparent page-left is not a substitute for the sitter's orientation. Named or numbered seats can make rotations distinct.

Source record: Bharatiya Pathshala — reviewed preparation lessons.

Arrangements & Grouping, explained simply

Try a small example

Core idea

Steps

Worked example

MAT: Intelligence and Critical Reasoning

CAT: Data Interpretation and Logical Reasoning

CMAT: Logical Reasoning

NMAT: Logical Reasoning

SNAP: Analytical and Logical Reasoning

Arrangements & Grouping, explained simply

Try a small example

Arrangements & Grouping: visual worked lesson

Core idea

Solving method

Worked example

Read the diagram

Exam-style application

Common mistakes

Core idea

Steps

Worked example

Common mistake

Circular Arrangements

Worked example

Grouping

Worked example

Application and strategy

Worked application

Common traps and strategy

Grouping with membership and capacity rules

Core idea

Solving method

Worked example

Common mistakes

Linear, circular and facing arrangements

Core idea

Solving method

Worked example

Common mistakes

MAT: Intelligence and Critical Reasoning

CAT: Data Interpretation and Logical Reasoning

CMAT: Logical Reasoning

NMAT: Logical Reasoning

SNAP: Analytical and Logical Reasoning

Arrangements & Grouping: visual worked lesson

Core idea

Solving method

Worked example

Read the diagram

Exam-style application

Common mistakes

Common mistake

Circular Arrangements

Worked example

Grouping

Worked example

Application and strategy

Worked application

Common traps and strategy

Grouping with membership and capacity rules

Core idea

Solving method

Worked example

Common mistakes

Linear, circular and facing arrangements

Core idea

Solving method

Worked example

Common mistakes