Permutations

ACT Statistics and Probability

In this lesson:

• Count ordered arrangements using factorials
• Apply the formula
• Solve ACT-style permutation problems

Your Learning Goals for This Lesson

By the end of this lesson, you will be able to:

1. Evaluate factorial expressions including and
2. Explain why order matters in permutations
3. Apply to count ordered arrangements
4. Solve applied permutation problems (rankings, codes, seating)

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

• (definition — prevents division by zero)
• (cancel )

Quick Check: Simplify a Factorial Expression

Cancel the common factor. Think before advancing…

When Order Matters: Permutation Situations

Key question: Does rearranging the selection change the outcome?

Deriving the Permutation Formula from Counting

Problem: Choose president, VP, and secretary from 20 club members.

• 20 choices for president
• × 19 remaining for VP
• × 18 remaining for secretary

= total items; = positions being filled

Example 1: Award Rankings —

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

• Order matters (1st 2nd)
• ,

Example 2: 4-Digit Code —

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

• Order matters (1234 4321)
• ,

Quick Check: Apply the Formula

Set up the formula and simplify. Think before advancing…

Special Case: Arranging All Items

When (arrange everything in the set):

Example: 5 books on a shelf →

prevents division-by-zero here.

ACT Language Cues for Permutation Problems

Phrases that signal a permutation:

• "arranged in order" → ordered positions
• "ranked / 1st, 2nd, 3rd" → distinct positions
• "no repeated digits" → pool shrinks with each pick
• "roles assigned" → distinct roles = order matters

Strategy: Underline the signal phrase first, then apply .

ACT Problem 1: Seating in a Row

5 students in 5 seats. How many arrangements?

• All 5 are being placed in distinct positions

Show the work:

ACT Problem 2: Arranging Letters of a Word

How many ways can the letters W, O, R, K be arranged?

• 4 distinct letters, 4 positions

Verify: 4 choices for first, 3 for second, 2 for third, 1 for last.

ACT Practice: Solve Three Permutation Problems

Solve before advancing:

1. A race has 6 runners. How many ways can gold, silver, bronze be awarded?
2. A 3-digit code uses digits 1–9, no repeats. How many codes exist?
3. How many ways can 4 different books fill 4 shelf spots?

Check Your Work: Practice Solutions

1. ways

2. codes

3. arrangements

Order matters in all three. Each selection reduces the pool.

Key Takeaways: Permutation Rules and Warnings

• = product down to 1; always
• — multiply the top factors
• Order matters → permutation; not → combination

: never write

Cancel before multiplying — never expand full factorials

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 .