Factorial Notation Revision Notes for Grade 12 NSC Matric Mathematics

Factorial Notation

What is factorial notation?

Factorial notation is a mathematical shorthand that represents the product of all positive integers from 1 up to a given number. This concept naturally arises when we need to count arrangements or orderings of objects.

Consider this scenario: if 8 athletes are competing in a 400m race, in how many different ways can all 8 places be filled? The first position can be filled by any of the 8 athletes. Once the first athlete finishes, there are 7 athletes left for second place. After second place is determined, 6 athletes remain for third place, and so on. This gives us the calculation: $8 × 7 × 6 × 5 × 4 × 3 × 2 × 1 = 40,320$ .

infoNote

This type of calculation appears so frequently in mathematics that we use factorial notation to represent it more efficiently, making complex counting problems much more manageable.

Definition and basic concept

Factorial notation uses an exclamation mark (!) after a number to represent the product of all positive integers from that number down to 1.

For any positive integer $n$ :

$n! = n × (n-1) × (n-2) × ... × 3 × 2 × 1$

The key rule to remember is that the total number of possible arrangements of n different objects is n!

Important special case

chatImportant

By mathematical definition: 0! = 1

This might seem unusual, but it's essential for many mathematical formulas to work correctly. Remember this special case as it often appears in exam questions.

Calculating factorials

Using a calculator

Modern calculators make factorial calculations straightforward. Most scientific calculators have a dedicated factorial function that can handle even large numbers efficiently.

The general process involves these steps:

Input the number you want to find the factorial of
Press SHIFT on your CASIO calculator or 2ndF on your SHARP calculator
Press x! on your CASIO or n! on your SHARP calculator
Press equals to calculate the answer

infoNote

For example, calculating $12!$ using these steps gives 479,001,600. For numbers larger than about 13!, most calculators will display the result in scientific notation.

Manual calculation for small numbers

For smaller factorials, manual calculation helps you understand the concept better and can be faster than using a calculator.

lightbulbExample

Manual Factorial Calculations

$3! = 3 × 2 × 1 = 6$
$4! = 4 × 3 × 2 × 1 = 24$
$5! = 5 × 4 × 3 × 2 × 1 = 120$

Simplifying factorial expressions

Many exam questions involve simplifying expressions containing factorials. The key strategy is to expand the factorials strategically and cancel common terms rather than calculating each factorial separately.

lightbulbExample

Worked Example 1: Simplifying factorial fractions

To simplify $\frac{8!}{4!}$ :

Step 1: Expand the factorials

$\frac{8!}{4!} = \frac{8 × 7 × 6 × 5 × 4 × 3 × 2 × 1}{4 × 3 × 2 × 1}$

Step 2: Cancel the common terms $(4 × 3 × 2 × 1)$

$= 8 × 7 × 6 × 5 = 1,680$

lightbulbExample

Worked Example 2: General factorial division

To simplify $\frac{n!}{(n-1)!}$ :

Step 1: Expand the factorials

$\frac{n!}{(n-1)!} = \frac{n × (n-1) × (n-2) × ... × 3 × 2 × 1}{(n-1) × (n-2) × ... × 3 × 2 × 1}$

Step 2: Cancel all terms except $n$

$= n$

This demonstrates that $\frac{n!}{(n-1)!} = n$ , which is a useful pattern to remember.

lightbulbExample

Worked Example 3: Special case verification

When $n = 1$ , we get $\frac{1!}{0!} = \frac{1}{1} = 1$ .

Since both $1! = 1$ and $0! = 1$ , our identity $\frac{n!}{(n-1)!} = n$ still holds, giving us 1.

Common factorial operations

Understanding these patterns will help you solve factorial problems more efficiently and recognise shortcuts during exams.

n!/(n-1)! = n (as shown above)
n!/((n-k)!) = n × (n-1) × (n-2) × ... × (n-k+1) (first k terms of n!)
(n!)² = n! × n! (factorial squared)
n! × m! ≠ (n×m)! (factorials don't multiply this way)

infoNote

These patterns are frequently tested in exams, so it's worth memorising them to save time during problem-solving. The first pattern is especially common and can help you simplify complex expressions quickly.

Exam tips and common mistakes

Understanding where students typically make errors can help you avoid these pitfalls and approach factorial problems with confidence.

chatImportant

Common exam traps:

Remember that 0! = 1, not 0
Don't confuse $n!$ with $n × n × n...$ (that would be $n^n$ )
When dividing factorials, always look for opportunities to cancel terms before calculating
Check your calculator is in the right mode when calculating large factorials

infoNote

Problem-solving approach:

Identify what type of factorial expression you have
Expand factorials strategically (don't always expand everything)
Cancel common terms where possible
Calculate the remaining terms
Check your answer makes sense

bookmarkSummary

Key Points to Remember:

Factorial notation n! means multiply all integers from n down to 1
0! = 1 by definition - this is crucial to remember
Use your calculator's factorial function for large numbers
When simplifying factorial fractions, cancel common terms before calculating
The pattern n!(n-1)! = n appears frequently in exams

Factorial Notation (Grade 12 NSC Matric Mathematics): Revision Notes