Permutations

ACT Statistics and Probability

In this lesson:

• Count ordered arrangements using factorials
• Apply the formula
• Solve mixed problems by reading for order

Your Learning Goals for This Lesson

You will be able to:

1. Solve applied ordered-arrangement problems (rankings, codes, seating, officer roles) with , including
2. Explain why order matters in a given arrangement problem and name its and
3. Evaluate factorial expressions, including , , and quotients like

Listing All Arrangements of Three Items

List them: ABC, ACB, BAC, BCA, CAB, CBA → 6 arrangements

• 3 choices for position 1
• × 2 remaining for position 2
• × 1 remaining for position 3
• =

This product is written (read "3 factorial").

Factorial Notation and Key Special Cases

• , ,
• (pattern: )
• (cancel )

: cancel, don't expand.

Quick Check: Simplify a Factorial Expression

Cancel the common factor. Think before advancing…

A Problem Too Large to List

How many ways can a president, VP, and secretary be chosen from 20 club members?

• Listing every outcome is not practical
• Does order matter? Yes, because swapping the president and the VP gives a different outcome

Start from the counting principle. Think before advancing…

Deriving the Permutation Formula Step by Step

1. Order matters, no repeats (else )
2. Name (pool) and (positions)
3. Multiply factors: , , ...
4. :

Example 1: Award Rankings —

8 students compete; 3 receive 1st, 2nd, 3rd place awards.

• Does order matter? Yes, because 1st place is a different award from 2nd
• ,

Example 2: 4-Digit Code —

Digits 0–9, no repeated digits. How many 4-digit codes exist?

• Does order matter? Yes, because 1234 and 4321 are different codes
• ,

Example 3: All 5 Books —

In how many ways can all 5 books be arranged on a shelf?

• Does order matter? Yes, because swapping two books gives a different shelf
• , (arranging all items)

Mixed Problem 1: Distinct Roles

Stem: 9 students fill the roles of chair, recorder, and treasurer. How many ways?

A. 729 B. 504 C. 362,880 D. 27

Order? Yes: swapping chair and recorder changes the outcome.

Formula: (B)

Mixed Problem 2: Arranging Everything

Stem: In how many ways can 6 different books be lined up on a shelf?

A. 36 B. 720 C. 120 D. 46,656

Order? Yes: swapping two books changes the shelf.

Formula: (B)

Mixed Problem 3: Evaluate a Factorial Expression

Stem: What is ?

A. 6 B. 7 C. 4 D. 1

Reasoning: (pattern: ); .

Compute: (B)

Key Takeaways: Permutation Rules and Warnings

• = product down to 1;
• — multiply the top factors
• Order matters, no repeats → ; all →

, not 0: , not 6

Cancel before multiplying

What You Will Learn Next

Next lesson builds directly on today's permutation skills:

• Combinations: when order does not matter
• Formula:
• Same context — different question

Permutations are always larger than (or equal to) combinations for the same and .