Arrangements in a Line — Permutations with Repetition & Restriction (9709 PS1) — Mathematics

Introduction

Arrangements in a line is one of the most frequently examined topics in 9709 Probability & Statistics 1. It extends the basic counting principle (which you met when studying permutations) to two richer — and more challenging — scenarios:

Repetition: some objects are identical, so naïvely counting all arrangements overcounts.
Restriction: certain objects must, or must not, occupy adjacent positions (or some other constraint is imposed).

Mastering these techniques is essential because they appear either as standalone questions or as the first step in calculating probabilities over arrangements.

Core Concept

Arrangements without any restriction

Arranging $n$ distinct objects in a line gives $n!$ arrangements. This is your baseline.

Arrangements with repeated objects

When a group of $n$ objects contains repeated items — say $n_1$ identical objects of one type, $n_2$ of another, …, $n_k$ of another — many permutations are visually identical. The number of distinct arrangements is:

\frac{n!}{n_1!\, n_2!\, \cdots\, n_k!}

Why? The $n_1!$ ways of internally rearranging the first group all look the same, so we divide them out. We do this for every repeated group.

Arrangements with the restriction "must be adjacent"

Treat the objects that must be together as a single block. This reduces the number of items to arrange. Then multiply by the number of ways to arrange the objects within the block.

\text{Total} = (\text{arrangements of blocks}) \times (\text{internal arrangements of the fixed block})

Arrangements with the restriction "must NOT be adjacent"

Use complementary counting:

\text{(No restriction)} - \text{(the two ARE adjacent)}

Two or more rows

Treat each row as a separate linear arrangement. Count the ways to:

Select (and arrange) the people in the first row.
Arrange the remaining people in the second row.

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