Question 1

What is a factorial and how is it calculated?

Accepted Answer

The factorial of a non-negative integer n, written n!, is the product of all positive integers from 1 to n: n! = n × (n−1) × (n−2) × … × 2 × 1. Examples: 0! = 1 (by convention); 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. The 0!=1 convention is not arbitrary — it makes the combinatorics formulas work correctly for edge cases like "choose 0 from n."

Question 2

What are the main real-world applications of factorials?

Accepted Answer

Factorials appear throughout mathematics and science: Permutations — the number of ways to arrange n distinct objects in order is n! (e.g. 5 books on a shelf = 5! = 120 arrangements). Combinations — choosing r items from n: C(n,r) = n! ÷ (r! × (n−r)!). Probability — poker hand probabilities use C(52,5) = 52!/(5!×47!). Calculus — Taylor series: eˣ = 1 + x + x²/2! + x³/3! + … Physics — Stirling's approximation (ln n! ≈ n ln n − n) is used in statistical mechanics and thermodynamics.

Question 3

Why do factorials grow so fast?

Accepted Answer

Factorials are a product of increasing multipliers, so they grow superexponentially — faster than any exponential function like 2ⁿ. Quick scale reference: 10! ≈ 3.6 million; 15! ≈ 1.3 trillion; 20! ≈ 2.4 × 10¹⁸; 25! ≈ 1.6 × 10²⁵; 52! ≈ 8 × 10⁶⁷ (the number of ways to shuffle a deck of cards — more than the estimated number of atoms in the observable universe, which is ~10⁸⁰). This explosive growth is why even 100! (about 9.3 × 10¹⁵⁷) requires arbitrary-precision arithmetic to represent exactly.

Question 4

What is the connection between factorials and permutations vs. combinations?

Accepted Answer

Permutations count ordered arrangements; combinations count unordered selections. Permutations (order matters): P(n,r) = n! ÷ (n−r)!. Example: arranging 3 of 5 letters = P(5,3) = 5!/2! = 60. Combinations (order does not matter): C(n,r) = n! ÷ (r! × (n−r)!). Example: choosing 3 from 5 = C(5,3) = 5!/(3!×2!) = 10. The relationship: C(n,r) = P(n,r) ÷ r! — since each combination corresponds to r! permutations of the same items. Lottery odds use combinations: C(49,6) = 49!/(6!×43!) ≈ 14 million (why winning is so rare).

Question 5

What is the gamma function and how does it extend factorials?

Accepted Answer

Factorials are only defined for non-negative integers, but the Gamma function Γ(n) extends the concept to real and complex numbers: Γ(n) = (n−1)! for positive integers, and Γ(1/2) = √π ≈ 1.772. This means you can compute "half-factorial": (1/2)! = Γ(3/2) = √π/2 ≈ 0.886. The Gamma function appears in probability distributions (chi-squared, beta, Student's t), physics (quantum mechanics, statistical mechanics), and complex analysis. It is one of the most important functions in advanced mathematics.

Factorial Calculator

Quick reference — small factorials

Everyday Uses

Counting arrangements

Where nPr and nCr come from

Password intuition

Puzzle solving

Shuffling a deck of cards

Seating plans

Frequently Asked Questions

What is a factorial and how is it calculated?

What are the main real-world applications of factorials?

Why do factorials grow so fast?

What is the connection between factorials and permutations vs. combinations?

What is the gamma function and how does it extend factorials?

Related Calculators

Quick reference — small factorials

Everyday Uses

Counting arrangements

Where nPr and nCr come from

Password intuition

Puzzle solving

Shuffling a deck of cards

Seating plans

Frequently Asked Questions

What is a factorial and how is it calculated?

What are the main real-world applications of factorials?

Why do factorials grow so fast?

What is the connection between factorials and permutations vs. combinations?

What is the gamma function and how does it extend factorials?