Can You Distribute Factorials? Math Rules Explained
Can You Distribute Factorials? The Direct Answer
No. You cannot distribute factorials over addition, subtraction, multiplication, or division. The factorial function doesn't work that way.
If you've been trying to simplify expressions like (3 + 4)! by splitting it into 3! + 4!, you're doing it wrong. The same goes for multiplication: (3 × 4)! is not equal to 3! × 4!.
Factorials are non-linear operations. They don't distribute because the operation fundamentally changes how numbers combine. There's no algebraic shortcut here—brute force calculation is your only option.
What a Factorial Actually Is
A factorial is the product of all positive integers from 1 to n.
Notation: n! means multiply every number from 1 to n together.
Examples:
- 5! = 5 × 4 × 3 × 2 × 1 = 120
- 4! = 4 × 3 × 2 × 1 = 24
- 3! = 3 × 2 × 1 = 6
- 2! = 2 × 1 = 2
- 1! = 1
- 0! = 1 (special case)
That's it. No shortcuts, no distribution rules. Just multiply sequential integers.
Why Distribution Fails: The Proof
Let's kill the myth with actual numbers.
Factorials Over Addition
Claim: (a + b)! = a! + b!
Test with a = 3, b = 4:
Left side: (3 + 4)! = 7! = 5040
Right side: 3! + 4! = 6 + 24 = 30
5040 ≠ 30. Case closed.
Factorials Over Multiplication
Claim: (a × b)! = a! × b!
Test with a = 3, b = 2:
Left side: (3 × 2)! = 6! = 720
Right side: 3! × 2! = 6 × 2 = 12
720 ≠ 12. Also false.
Factorials Over Division
Claim: (a ÷ b)! = a! ÷ b!
Test with a = 6, b = 2:
Left side: (6 ÷ 2)! = 3! = 6
Right side: 6! ÷ 2! = 720 ÷ 2 = 360
6 ≠ 360. No distribution here either.
What Operations DO Work with Factorials
Factorials aren't completely isolated. There are legitimate ways they interact.
The Recursive Definition
This is the only "distribution" that works:
n! = n × (n − 1)!
That's not distribution—that's definition. It only works in one direction with the immediately preceding integer.
Examples:
- 7! = 7 × 6!
- 5! = 5 × 4!
- 10! = 10 × 9!
Factorials in Ratios (Cancellations)
You can cancel factorials in fractions:
n! ÷ (n − 1)! = n
Example: 8! ÷ 7! = 8
This works because:
8! ÷ 7! = (8 × 7 × 6 × ... × 1) ÷ (7 × 6 × ... × 1) = 8
The Ratio Pattern
When you have n! divided by (n − k)!, you get a product:
n! ÷ (n − k)! = n × (n − 1) × ... × (n − k + 1)
Example: 10! ÷ 7! = 10 × 9 × 8 = 720
Factorial Distribution: What You Can and Cannot Do
| Operation | Claim | Is It Valid? | Counterexample |
|---|---|---|---|
| Addition | (a + b)! = a! + b! | ❌ No | (1+2)! = 6, 1!+2! = 3 |
| Subtraction | (a − b)! = a! − b! | ❌ No | (5−2)! = 6, 5!−2! = 118 |
| Multiplication | (a × b)! = a! × b! | ❌ No | (3×2)! = 720, 3!×2! = 12 |
| Division | (a ÷ b)! = a! ÷ b! | ❌ No | (6÷2)! = 6, 6!÷2! = 360 |
| Recursive | n! = n × (n−1)! | ✅ Yes | Definition, always true |
| Ratio Cancellation | n! ÷ (n−1)! = n | ✅ Yes | Definition, always true |
How to Work with Factorial Expressions
When you encounter factorials in problems, here's what actually works:
Step 1: Expand or Simplify Based on Context
If you need a numerical answer, just calculate:
- 5! = 120
- 7! = 5040
- 10! = 3,628,800
If you have a ratio, cancel before multiplying:
Problem: 8! ÷ 4!
Don't do: 40320 ÷ 24 = 1680
Do: 8 × 7 × 6 × 5 = 1680
The second method is faster and avoids large number multiplication.
Step 2: Use the Ratio Rule for Combinations and Permutations
These formulas are where factorial ratios appear most:
Permutations: P(n,r) = n! ÷ (n − r)!
Combinations: C(n,r) = n! ÷ [r! × (n − r)!]
Always cancel before calculating to keep numbers manageable.
Example: C(10,3) = 10! ÷ (3! × 7!)
= (10 × 9 × 8) ÷ (3 × 2 × 1) = 720 ÷ 6 = 120
Step 3: Know When to Stop
Factorials grow absurdly fast. By 13!, you're past 6 billion. By 20!, you're in the quintillions.
Most textbook problems involving factorials will keep n small (typically n ≤ 10) unless you're using the ratio rules to avoid big numbers.
Common Mistakes to Avoid
- Splitting addition: (a+b)! ≠ a! + b! — calculate the sum first, then factorial
- Splitting multiplication: (a×b)! ≠ a! × b! — calculate the product first, then factorial
- Assuming linearity: Factorials are not linear. They don't distribute, combine, or simplify across operations
- Forgetting 0!: 0! = 1, not 0. This matters in combinations
The Bottom Line
Factorials don't distribute. There's no algebraic trick that lets you break n! into smaller factorials combined with basic operations.
Your options:
- Calculate directly: n! = 1 × 2 × ... × n
- Use the recursive property: n! = n × (n−1)!
- Cancel in ratios: n! ÷ (n−k)! = product of k terms
That's it. No shortcuts. No distribution. Just multiplication and cancellation.