Factorial Calculator

Calculation Type

n!

Factorial

n!!

Double Factorial

!n

Subfactorial

Γ(n)

Gamma Function

n - Input Value

CalculateClear

Common Factorial Values

0!

1

1!

1

2!

2

3!

6

4!

24

5!

120

6!

720

7!

5,040

8!

40,320

9!

362,880

10!

3,628,800

12!

479,001,600

Reference

Factorial (n!): Product of all positive integers up to n.
n! = n × (n-1) × (n-2) × ... × 1

Double Factorial (n!!): Product of integers with same parity.
n!! = n × (n-2) × (n-4) × ...

Subfactorial (!n): Number of derangements (permutations with no fixed points).
!n = n! × Σ(-1)^k/k! for k=0 to n

Gamma Function Γ(n): Extension of factorial to real numbers.
Γ(n) = (n-1)! for positive integers

How to Use the Factorial Calculator

This calculator computes factorials and related functions including double factorial, subfactorial (derangements), and the gamma function.

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Factorial (n!)

The factorial of n is the product of all positive integers from 1 to n.

Definition: n! = n × (n-1) × (n-2) × … × 2 × 1

Special cases:

• 0! = 1 (by convention)
• 1! = 1

Common Factorial Values

n	n!
0	1
1	1
2	2
3	6
4	24
5	120
6	720
7	5,040
8	40,320
9	362,880
10	3,628,800
12	479,001,600
15	1,307,674,368,000
20	2,432,902,008,176,640,000

Applications

• Counting arrangements (permutations)
• Binomial coefficients
• Probability distributions
• Taylor series

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Double Factorial (n!!)

The double factorial multiplies every other integer from n down to 1 or 2.

Definition:

• For odd n: n!! = n × (n-2) × (n-4) × … × 3 × 1
• For even n: n!! = n × (n-2) × (n-4) × … × 4 × 2

Special cases:

• 0!! = 1
• 1!! = 1
• (-1)!! = 1

Examples

n	n!!	Expansion
1	1	1
2	2	2
3	3	3 × 1
4	8	4 × 2
5	15	5 × 3 × 1
6	48	6 × 4 × 2
7	105	7 × 5 × 3 × 1
8	384	8 × 6 × 4 × 2
9	945	9 × 7 × 5 × 3 × 1
10	3,840	10 × 8 × 6 × 4 × 2

Relationship to Factorial

• For even n: n!! = 2^(n/2) × (n/2)!
• For odd n: n!! = n! / (n-1)!!

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Subfactorial (!n) - Derangements

The subfactorial counts derangements: permutations where no element stays in its original position.

Definition: !n = n! × Σ((-1)^k / k!) for k = 0 to n

Recurrence: !n = (n-1) × (!(n-1) + !(n-2))

Examples

n	!n	Probability
0	1	100%
1	0	0%
2	1	50%
3	2	33.3%
4	9	37.5%
5	44	36.7%
6	265	36.8%
7	1,854	36.8%
8	14,833	36.8%
10	1,334,961	36.8%

Note: As n increases, the probability of a derangement approaches 1/e ≈ 36.79%

Applications

• Secret Santa problem
• Hat check problem
• Card shuffling analysis

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Gamma Function Γ(n)

The gamma function extends the factorial to non-integer values.

Key property: Γ(n) = (n-1)! for positive integers

Relationship: Γ(n+1) = n × Γ(n)

Special Values

x	Γ(x)
1	1
2	1
3	2
4	6
5	24
1/2	√π ≈ 1.7725
3/2	√π/2 ≈ 0.8862
-1/2	-2√π ≈ -3.5449

Notable Formula

Γ(1/2) = √π

This means (−1/2)! = √π, extending factorials to half-integers!

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Stirling’s Approximation

For large n, factorial can be approximated:

n! ≈ √(2πn) × (n/e)^n

n	n!	Stirling Approx	Error
5	120	118.02	1.7%
10	3,628,800	3,598,696	0.8%
20	2.43 × 10^18	2.42 × 10^18	0.4%

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Factorial Growth

Factorials grow extremely fast—faster than exponential!

n	n!	2^n	n^n
5	120	32	3,125
10	3,628,800	1,024	10^10
20	2.43 × 10^18	1.05 × 10^6	10^26

Number of digits in n!: approximately n × log₁₀(n/e) + 0.5 × log₁₀(2πn)

• 100! has 158 digits
• 1000! has 2,568 digits

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Related Resources

• Combinations Calculator - Uses factorials
• Counting Principles - Fundamentals
• Binomial Distribution - Probability application