Factorial Notation: Concepts, Formulas and Examples

Factorial notation writes the product 1 × 2 × 3 × … × n as n!, read as "n factorial". For example, 5! = 1 × 2 × 3 × 4 × 5 = 120. It is the standard shorthand for the "multiply all numbers down to 1" operation that appears in every counting formula.

This matters because permutations, combinations, and probability expressions all use factorials. Writing 52! is far cleaner than writing the whole product, and the recursive property n! = n × (n − 1)! lets large factorials cancel neatly in fractions.

Key Concepts

• n!: defined for non-negative integers as n! = 1 × 2 × … × n, with the special value 0! = 1.
• Recursive property: n! = n × (n − 1)!, which lets you peel off factors one at a time.
• Cancellation: in a fraction like 7!/4!, the 4! cancels, leaving 7 × 6 × 5.
• Growth: factorials grow extremely fast — 10! = 3,628,800.

Formulas

• Recursion: n! = n × (n − 1)!, for every integer n ≥ 1.
• Partial product: n! / (n − r)! = n × (n − 1) × … × (n − r + 1), a product of r terms.
• Combination: C(n, r) = n! / (r! (n − r)!), defined for integers 0 ≤ r ≤ n.
• Base values: 0! = 1 and 1! = 1.

Worked Example

Simplify (5! + 4!) / 3!. First compute the factorials: 5! = 120 and 4! = 24, so the numerator is 120 + 24 = 144. The denominator is 3! = 6. Dividing: 144 / 6 = 24. As a second check of cancellation, 7! / 4! = (7 × 6 × 5 × 4!) / 4! = 7 × 6 × 5 = 210 — the 4! cancels completely.

Common Misconceptions

• "0! = 0." Wrong — 0! is defined as 1. This definition makes formulas like C(n, 0) = 1 work correctly.
• "(m + n)! = m! + n!." Wrong — factorials do not distribute over addition. For example, (2 + 3)! = 120, but 2! + 3! = 8.
• "(−3)! exists." Wrong — factorial is defined only for non-negative integers in this course.

Real-World Applications

• Permutations and combinations: every arrangement formula is built from factorials.
• Probability: computing the number of possible outcomes in card games and lotteries.
• Statistics: the binomial distribution and many test statistics use factorials.
• Computer science: analysing the running time of algorithms that try all orderings.

Factorial notation compresses long products into a single symbol and cancels beautifully in fractions. Remember 0! = 1, use n! = n × (n − 1)! to simplify, and you have the building block of all counting mathematics.