Chapter 10: Permutations and Arrangements

Order-based counting, slot methods, codes, circular seating, and flippable ring arrangements.

Chapter Overview

Medal StandOrdered awards and slot multiplication. Lock CodesCodes with and without repeated digits. Campfire SeatingCircular arrangements. Keyring PermutationsFlippable rings. Inseparable TeammatesBlock method.

A permutation problem counts ordered outcomes. If changing the position of an object changes the result, the problem belongs in this chapter rather than the combinations chapter.

  • The Fundamental Counting Principle fills ordered slots by multiplication.
  • Code problems split into repetition allowed and repetition forbidden.
  • The formula $^nP_r$ applies when only part of a set is selected and arranged.
  • Circular arrangement logic handles campfires, round tables, and necklace-style rotations.
  • Flippable rings divide by 2 when mirror images are physically identical.

10.1 Fundamental Counting Principle

The Fundamental Counting Principle says that when a process is built from several independent stages, multiply the number of choices at each stage.

$$N=a_1a_2a_3\cdots a_k$$
Core Concept

When a task is built from ordered stages, the counting principle tells us to multiply the number of choices in each slot.

Worked Example 10.1.1 — Medal Stand

A school assigns three distinct awards: gold, silver, and bronze. There are twelve finalists, and no finalist can receive more than one award. How many award arrangements are possible?

  • The awards are distinct positions, so order matters.gold, silver, bronze
  • The choices decrease as finalists are used.12 \cdot 11 \cdot 10
$$12\cdot11\cdot10=1320$$

10.2 Permutation Formula

The permutation formula is a compact version of the slot method when $r$ ordered positions are filled from $n$ available objects without repetition.

$$^nP_r=\frac{n!}{(n-r)!}$$
Core Concept

A permutation is a slot count with no repetition; the unused tail of the factorial cancels away.

  • $n$ is the number of available objects.
  • $r$ is the number of ordered positions being filled.
  • $(n-r)!$ cancels the unused part of the full factorial.

10.3 Codes and Locks

Code problems are ordered because the position of each digit or letter matters. The central question is whether repetition is allowed.

Worked Example 10.3.1 — Lock Without Repeats

A four-digit lock uses digits 0 through 9 and does not allow any digit to repeat. How many lock outcomes are possible?

  • There are 10 choices for the first digit.
  • After a digit is used, it cannot be used again, so the choices decrease.
$$10\cdot9\cdot8\cdot7=5040$$

Worked Example 10.3.2 — Keypad With Repeats

A keypad code allows repeated digits. A code has four digits. How many possible keypad codes exist?

  • Each digit position has 10 choices.
  • Because repetition is allowed, the number of choices does not decrease.
$$10^4=10000$$

10.4 Circular Arrangements

In a circle, rotations of the same seating are not new arrangements. The standard method is to fix one person as a reference point, then arrange everyone else.

$$\text{Circular arrangements of }n\text{ distinct objects}=(n-1)!$$
Core Concept

For circular arrangements, subtract one from the number of objects and count (n-1)!.

Worked Example 10.4.1 — Campfire Seating

Nine campers sit around a campfire. If two seatings that differ only by rotation are considered the same, how many campfire arrangements are possible?

  • Fix one camper to remove rotational duplicates.9 \text{ campers} \rightarrow 1 \text{ fixed reference}
  • Arrange the remaining campers around the fixed point.(9-1)! = 8!
$$8!=40320$$

10.5 Flippable Rings and Keyrings

A keyring can be turned over. That means a clockwise order and its mirror-image counterclockwise order describe the same physical ring.

$$\text{Flippable ring arrangements}=\frac{(n-1)!}{2}$$
Core Concept

For a flippable ring, first remove rotations with $(n-1)!$, then divide by 2 because each mirror image is the same physical arrangement.

Worked Example 10.5.1 — Keyring Permutations

A keyring holds eight unique keys. Two arrangements are considered the same if the ring can be rotated or flipped over to match. How many distinct keyrings are possible?

  • First remove rotational duplicates.(8-1)! = 7!
  • Then divide by 2 because each clockwise order has the same flipped version.\frac{7!}{2}
$$\frac{(8-1)!}{2}=\frac{7!}{2}=2520$$

10.6 Inseparable Teammates

Some arrangement problems say that two people or objects must stay together. The standard move is to bind the inseparable group into one block, arrange the blocks, then arrange the members inside the block.

Core Concept

When a group must stay together, treat that group as one block first, then multiply by the internal arrangements of the block.

Worked Example 10.6.1 — Inseparable Teammates

Six students line up for a photograph. Ava and Ben must stand next to each other. How many lineups are possible?

  • Treat Ava-and-Ben as one block.[AB], C, D, E, F \rightarrow 5 \text{ objects}
  • Arrange the 5 objects.5!
  • Arrange the teammates inside the block.2!
$$5!\cdot2!=120\cdot2=240$$

10.7 Permutation Calculator

This calculator fills the permutation formula so the slot count and factorial form stay visible.