4.1.2

Factorials

Test yourself

Factorial Notation

Factorial notation is used as shorthand for certain products of numbers.

Illustrative background for FactorialsIllustrative background for Factorials ?? "content

Factorials

  • The factorial of a number is equal to the number multiplied by all of the natural numbers smaller than it:
    • For example, 4 factorial is equal to 4 × 3 × 2 × 1 = 24.
Illustrative background for Factorial notationIllustrative background for Factorial notation ?? "content

Factorial notation

  • Factorial notation makes writing factorials easier.
  • The factorial of a whole number nn is written as n!n!.
  • 6 factorial = 6!=6×5×4×3×2×1=7206! = 6\times5\times4\times3\times2\times1 = 720
Illustrative background for FactorialsIllustrative background for Factorials ?? "content

Factorials

  • The factorial of 0 is defined to be equal to 1.
Illustrative background for ApplicationsIllustrative background for Applications ?? "content

Applications

  • Factorials are used in many areas of maths. One common use of factorials is counting the number of ways to pick or arrange objects.
  • For example, if you have 3 unique items, the number of ways to arrange these items is equal to 3!=3×2×1=63!=3\times2\times1=6.
Illustrative background for ApplicationsIllustrative background for Applications ?? "content

Applications

  • We can use factorials to work out the number of ways to pick rr distinct objects from a collection of nn objects.
  • This number is called "nn choose rr" and is written as nCr^{n}C_{r} or (nr)n\choose r:
    • (nr)=n!r!(nr)!{n\choose r} = \frac{\small n!}{\small r!(n-r)!}
Illustrative background for ExampleIllustrative background for Example ?? "content

Example

  • A restaurant offers five side dish options. Your meal comes with two side dishes.
    • How many ways can you select your side dishes?
  • The total number of options n=5n=5, and r=2r=2, so we work out (52)5\choose 2:
    • (52)=5!2!(53)!=5!2!3!=1202×6=10{5\choose 2} = \frac{\small 5!}{\small 2!(5-3)!}= \frac{\small 5!}{\small 2!3!} = \frac{\small 120}{\small 2\times6} = 10
Illustrative background for Connection to Pascal's triangleIllustrative background for Connection to Pascal's triangle ?? "content

Connection to Pascal's triangle

  • We can work out each number in each row of Pascal's triangle using (nr)n\choose r.
  • The rrth number in the nnth row of Pascal's triangle is equal to:
    • n1Cr1=(n1r1)=(n1)!(r1)!((n1)(r1))!=(n1)!(r1)!(nr)!^{n-1}C_{r-1}={n-1\choose r-1} = \frac{(n-1)!}{(r-1)!((n-1)-(r-1))!}=\frac{(n-1)!}{(r-1)!(n-r)!}
Illustrative background for ExampleIllustrative background for Example ?? "content

Example

  • The 3rd number of the 4th row in Pascal's triangle is equal to:
    • 3C2=(32)=3!2!1!=62=3^{3}C_{2}={3\choose2} = \frac{3!}{2!1!} = \frac{6}{2} = 3

Jump to other topics

1Proof

2Algebra & Functions

2.1Powers & Roots

2.2Quadratic Equations

2.3Inequalities

2.4Polynomials

2.5Graphs

2.6Functions

2.7Transformation of Graphs

2.8Partial Fractions (A2 Only)

3Coordinate Geometry

4Sequences & Series

5Trigonometry

6Exponentials & Logarithms

7Differentiation

8Integration

9Numerical Methods

10Vectors

Go student ad image

Unlock your full potential with GoStudent tutoring

  • Affordable 1:1 tutoring from the comfort of your home

  • Tutors are matched to your specific learning needs

  • 30+ school subjects covered

Book a free trial lesson