Why does 0! equal 1 instead of 0?

In combinatorics, n! counts the number of ways to arrange n items. There is exactly 1 way to arrange an empty set of 0 items. Also, the formula n! = (n+1)! / (n+1) means 0! = 1! / 1 = 1.

Can you calculate the factorial of a negative number?

For standard integers, factorials are only defined for non-negative integers (0, 1, 2, 3...). In advanced calculus, the Gamma function extends factorials to complex numbers with poles at negative integers.

How do I print infinite new factorial worksheets?

Click "Unlimited & Shuffle" to generate fresh randomized factorial worksheets with customizable difficulty and problem counts.

🎓 Grades 6-9📄 Free PDF + Keys♾️ Dynamic Generator

FactorialFREEWorksheet Generator(Grades 6-9)

Factorial worksheets with step-by-step answer keys. Generate unlimited unique practice sheets with fresh problems for Grades 6–9.

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Set A • 10 Qs

PanMaths.comFactorial Worksheets • Set A (Medium)

Name: __________________________

Date: _____________ Period: ____

Score: _____ / 10

5!= ______

Workspace:

7!= ______

Workspace:

8!= ______

Workspace:

3!= ______

Workspace:

9!= ______

Workspace:

1!= ______

Workspace:

4!= ______

Workspace:

6!= ______

Workspace:

2!= ______

Workspace:

10.

5!= ______

Workspace:

PanMaths.com • Free Math Worksheet Generator • Set A (Medium)Page 1 of 2

PanMaths.comAnswer Key & Worked Solutions • Set A (Medium)

Answer Key

Total Problems: 10

5!= 120

1. Expand Factorial Chain: 5! = 5 × 4 × 3 × 2 × 1

2. Evaluate Product: 120

3. Value: 5! = 120 ✅

7!= 5040

1. Expand Factorial Chain: 7! = 7 × 6 × 5 × 4 × 3 × 2 × 1

2. Evaluate Product: 5,040

3. Value: 7! = 5,040 ✅

8!= 40320

1. Expand Factorial Chain: 8! = 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1

2. Evaluate Product: 40,320

3. Value: 8! = 40,320 ✅

3!= 6

1. Expand Factorial Chain: 3! = 3 × 2 × 1

2. Evaluate Product: 6

3. Value: 3! = 6 ✅

9!= 362880

1. Expand Factorial Chain: 9! = 9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1

2. Evaluate Product: 362,880

3. Value: 9! = 362,880 ✅

1!= 1

1. Product Chain: 1! = 1.

2. Evaluation: 1! = 1 ✅

4!= 24

1. Expand Factorial Chain: 4! = 4 × 3 × 2 × 1

2. Evaluate Product: 24

3. Value: 4! = 24 ✅

6!= 720

1. Expand Factorial Chain: 6! = 6 × 5 × 4 × 3 × 2 × 1

2. Evaluate Product: 720

3. Value: 6! = 720 ✅

2!= 2

1. Expand Factorial Chain: 2! = 2 × 1

2. Evaluate Product: 2

3. Value: 2! = 2 ✅

10.

5!= 120

1. Expand Factorial Chain: 5! = 5 × 4 × 3 × 2 × 1

2. Evaluate Product: 120

3. Value: 5! = 120 ✅

PanMaths.com • Answer Key & Worked SolutionsPage 2 of 2

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Worksheet Info

Categoryarithmetic

Grade LevelGrades 6-9

DifficultyEasy to Hard

Time15-20 min

Skills Practiced

✓ Sequential multiplication
✓ Growth rate estimation
✓ Math logic

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GCF & LCM Number Snake Arcade

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Mastering the Exclamation Point

A factorial is simply the product of an integer and all the integers below it. While the concept is simple, the numbers grow incredibly fast. For example, while 3! is only 6, 10! is over 3.6 million!

Our worksheets help students get comfortable with this rapid growth by providing tiered problem sets from easy (n < 5) to advanced (n > 10).

Classroom & Home Use

Teachers can use these sheets for timed drills or as part of a counting unit. Parents can use them to reinforce multiplication skills, as calculating a factorial is essentially a long string of multiplication operations.

💡 1. What is a Factorial?

The factorial of a positive integer n (written n!) is the product of all positive integers less than or equal to n: 5! = 5 × 4 × 3 × 2 × 1 = 120.

💡 2. The Special Zero Factorial Rule: 0! = 1

By mathematical convention and combinatorics definition, 0! = 1 (there is exactly 1 way to arrange zero items).

💡 3. Fast Step-by-Step Multiplication Tip

Group numbers that make round tens: 5! = (5 × 2) × (4 × 3) × 1 = 10 × 12 = 120!

Worksheet Facts by Grade Level

Grade Level	Factorial Standard	Target Numbers
Grade 5–6	Understand n! notation and evaluate small descending products	0! to 5! (Values 1 to 120)
Grade 7–8	Evaluate multi-step chains and simplify factorial fractions (CCSS.7.SP.C.8)	2! to 8! (Values up to 40,320)
High School Algebra / Stats	Use factorials in permutations and combinations formulas (CCSS.HSS.CP.B.9)	Up to 10! and beyond

Why Factorials are the Foundation of Counting & Probability

In mathematics, Factorials (n!) represent the total number of ways to arrange or permute $n$ distinct objects in a straight line.

For example, 4 students can line up in $4! = 4 \times 3 \times 2 \times 1 = 24$ unique orders. Factorials form the operational core of permutations ($P(n, r)$), combinations ($C(n, r)$), and binomial probability distributions.

First 10 Factorials Reference Table

0! = 1
1! = 1
2! = 2 × 1 = 2
3! = 3 × 2 × 1 = 6
4! = 4 × 3 × 2 × 1 = 24
5! = 5 × 4 × 3 × 2 × 1 = 120
6! = 6 × 120 = 720
7! = 7 × 720 = 5,040
8! = 8 × 5,040 = 40,320
9! = 9 × 40,320 = 362,880
10! = 10 × 362,880 = 3,628,800

Recursive Property of Factorials

Every factorial contains the previous factorial inside it: n! = n × (n - 1)!. This recursive property allows fast simplification in fraction equations like $\frac{8!}{6!} = 8 \times 7 = 56$.

Benefits of Mastery

🎲 Probability & Combinatorics

Prepares students for high school statistics, permutations, and combinations.

⚡ Rapid Growth Intuition

Teaches students how super-exponential functions grow compared to linear and polynomial math.

🧠 Mental Arithmetic Power

Builds associative grouping strategies when multiplying multi-digit descending chains.

🖨️ Clean Printable Answer Keys

Includes complete product chain expansions on every verified solution sheet.