Factorial

College/University

Definition

It is the product of an integer with the rest of its smaller positive integers. A factorial is represented by a "!" sign after a number. It indicates that you should multiply a series of descending numbers. For example, 5! (read as "5 factorial") means you should carry out: 5 x 4 x 3 x 2 x 1 = 120.

Worked examples

\(5! = 5 \times 4 \times 3 \times 2 \times 1 = 120\)

Multiply 5 by every positive integer below it, all the way down to 1.

\(0! = 1\)

Zero factorial is defined to be 1, a special case that makes many formulas work.

\(\frac{7!}{5!} = \frac{7 \times 6 \times 5!}{5!} = 7 \times 6 = 42\)

Cancel the common factorial; only the top factors remain.

Common mistakes

\(5! = 5 \times 4 = 20\) → \(5! = 5 \times 4 \times 3 \times 2 \times 1 = 120\) You must multiply all integers down to 1, not just the first two.
\(0! = 0\) → \(0! = 1\) Zero factorial is defined as 1 by convention, not 0.
\((-3)!\) is defined → Factorials are only defined for non-negative integers Negative numbers do not have standard factorials.

Where you'll use it next

You'll use factorials in permutations and combinations to count arrangements and selections, in binomial expansions, probability problems, and Taylor series in calculus.

Found in 1 StudyPug lesson

Factorial notation

Grade 12 Math

12th Grade12th