Probability and Counting Principles: Background You’ll Need 2

  • Understand what factorials are and calculate them for whole numbers

Factorials

Factorials are a fundamental concept in mathematics, playing a crucial role in combinatorics, probability theory, and various other mathematical fields. The factorial of a non-negative integer [latex]n[/latex], denoted as [latex]n![/latex], is the product of all positive integers less than or equal to [latex]n[/latex].

factorial

For any non-negative integer [latex]n[/latex], the factorial of [latex]n[/latex] is defined as:

[latex]n! = n \times (n-1) \times (n-2) \times \cdots \times 3 \times 2 \times 1[/latex]

Special cases:

  • [latex]0! = 1[/latex] (by definition)
  • [latex]1! = 1[/latex]

Let’s compute the factorials for the first few natural numbers:

  1. [latex]1! = 1[/latex]
  2. [latex]2! = 2 \times 1 = 2[/latex]
  3. [latex]3! = 3 \times 2 \times 1 = 6[/latex]
  4. [latex]4! = 4 \times 3 \times 2 \times 1 = 24[/latex]
  5. [latex]5! = 5 \times 4 \times 3 \times 2 \times 1 = 120[/latex]
Factorials grow extremely quickly. Even for relatively small values of [latex]n[/latex], [latex]n![/latex] becomes very large. For example, [latex]10! = 3,628,800[/latex].
Compute [latex]7![/latex].

For any positive integers [latex]m[/latex] and [latex]n[/latex] where [latex]m < n[/latex], [latex]n![/latex] is always divisible by [latex]m![/latex].

When you have a fraction with factorials, look for a matching factorial pattern in both the numerator and denominator. Just like canceling regular numbers, you can cancel the smaller factorial by dividing both top and bottom by it. For example, if you see [latex]\frac{8!}{6!}[/latex], you can rewrite [latex]8![/latex] as [latex]8 \cdot 7 \cdot 6![/latex] and then cancel the [latex]6![/latex]:

[latex]\frac{8!}{6!} = \frac{8 \cdot 7 \cdot 6!}{6!} = 8 \cdot 7 = 56[/latex]