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Factorials

Maths: Edexcel A Level Pure Maths, topic 4.1.2

Factorial Notation

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

Factorials

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.
Factorial notation

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
Factorials

Factorials

  • The factorial of 0 is defined to be equal to 1.
Applications

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.
Applications

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!(n−r)!{n\choose r} = \frac{\small n!}{\small r!(n-r)!}
Example

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!(5−3)!=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
Connection to Pascal's triangle

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:
  • n−1Cr−1=(n−1r−1)=(n−1)!(r−1)!((n−1)−(r−1))!=(n−1)!(r−1)!(n−r)!^{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)!}
Example

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

Test yourself on Factorials

  • What is the correct way to write 3 factorial?

  • What is the correct equation for nn choose rr?

  • What is 25!23!×2!\frac{\small 25!}{\small 23!\times2!} equivalent to?

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