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Fundamentals of Business Mathematics and Statistics · Permutation and Combinations

Factorial Notation and Its Properties Explained

Updated 10 October 2026 · Fact-checked

The factorial of a natural number n, written n!, is the product of all whole numbers from 1 to n. So 5! = 5 × 4 × 3 × 2 × 1 = 120. By definition 0! = 1. To simplify expressions, expand the larger factorial only until it shows the smaller one, then cancel.

Understand Factorial Notation and Its Properties

Factorial is a short way to write a long multiplication. Instead of writing 6 × 5 × 4 × 3 × 2 × 1, you write 6! and read it as "six factorial". It is defined only for whole numbers: 0, 1, 2, 3 and so on.

The key idea is that a factorial contains smaller factorials inside it. 6! = 6 × 5!, and 5! = 5 × 4!. In general, n! = n × (n − 1)!. This one property is behind almost every simplification question.

Why is 0! = 1? Use the same property backwards. Since n! = n × (n − 1)!, put n = 1. Then 1! = 1 × 0!. As 1! = 1, we get 0! = 1. It is a definition chosen so the rule keeps working. It also fits counting: there is exactly one way to arrange zero objects, which is to do nothing.

Factorials grow very fast. 5! = 120, 6! = 720, 7! = 5040. Do not try to compute big factorials in full. Cancel common parts instead. Factorials are the base of permutations and combinations, so you need this topic to be quick and accurate.

Key formulas to remember

Definition of n!
n! = n × (n − 1) × (n − 2) × ... × 3 × 2 × 1
Valid for natural numbers n. Example: 4! = 24.
Recursive property
n! = n × (n − 1)!
Use it to expand a factorial just enough to cancel. Also valid as n! = n(n − 1)(n − 2)!, for n ≥ 2.
Zero factorial
0! = 1
It is a definition. Also 1! = 1.
Values to remember
1! = 1, 2! = 2, 3! = 6, 4! = 24, 5! = 120, 6! = 720, 7! = 5040
Knowing these saves time in MCQs.
Ratio of factorials
n! ÷ r! = n × (n − 1) × ... × (r + 1), for n > r
Cancel the smaller factorial; the product has n − r terms.
Not distributive
(m + n)! ≠ m! + n! and (m × n)! ≠ m! × n!
Always compute the bracket first, then take the factorial.

How to solve Factorial Notation and Its Properties questions

Use this method for any question that asks you to evaluate, simplify or solve an equation with factorials.

  1. 1Check that every number inside a factorial is a whole number. Factorials of negatives or fractions are not defined.
  2. 2Work out any bracket first, such as (n − 2)! or (8 − 5)!. Never split a factorial over addition or multiplication.
  3. 3Find the smaller factorial in the expression and expand the larger one down to it using n! = n × (n − 1)!.
  4. 4Cancel the common factorial from numerator and denominator.
  5. 5Multiply or simplify the few terms that remain.
  6. 6If there is an unknown n, form an ordinary equation from the remaining terms and solve it. Reject any answer that is negative or not a whole number.
  7. 7Substitute your answer back, or check it against the options, before marking.

Quickest way: Expand only to the smaller factorial

When to use it: Use for fractions with factorials in both numerator and denominator, and for equations like n! = 20 × (n − 2)!.

  1. Identify the smallest factorial present.
  2. Write the biggest factorial as a product ending in that smallest factorial.
  3. Cancel it at once.
  4. Solve the short product or the small equation left.
  5. In MCQs, test the options with known values (5! = 120, 6! = 720) when the equation looks messy.

Common mistakes in Factorial Notation and Its Properties

  • Taking 0! = 0

    Zero times anything is zero, so it feels natural.

    Fix: Remember 0! = 1 by definition. It comes from 1! = 1 × 0!.

  • Writing (m + n)! = m! + n!

    Students treat the factorial sign like a bracket that distributes.

    Fix: Add inside the bracket first. For example, (2 + 3)! = 5! = 120, but 2! + 3! = 8.

  • Writing 8! ÷ 4! = 2!

    Students divide the numbers 8 and 4 and keep the factorial sign.

    Fix: Expand: 8! ÷ 4! = 8 × 7 × 6 × 5 = 1680.

  • Expanding too far or computing the full value

    Students do not look for the common factorial to cancel.

    Fix: Stop expanding at the smaller factorial and cancel it.

  • Writing (n − 2)! as n! − 2!

    Confusing a bracket with a subtraction of factorials.

    Fix: Treat (n − 2)! as one unit. Use n! = n(n − 1)(n − 2)!.

  • Accepting a negative or fractional n

    Students stop after solving the quadratic.

    Fix: Factorial needs a whole number n. Reject other roots.

Worked examples

Example 1

Simplify 9! ÷ (6! × 3!). Choose from: (A) 14 (B) 28 (C) 84 (D) 504

Show the solution
  1. The smallest large factorial in the denominator is 6!, so expand 9! down to 6!.
  2. 9! = 9 × 8 × 7 × 6!.
  3. So 9! ÷ (6! × 3!) = (9 × 8 × 7 × 6!) ÷ (6! × 3!).
  4. Cancel 6!: (9 × 8 × 7) ÷ 3!.
  5. 9 × 8 × 7 = 504 and 3! = 6.
  6. 504 ÷ 6 = 84.

Answer: 84 (option C)

Example 2

If n! = 30 × (n − 2)!, find n. Choose from: (A) 5 (B) 6 (C) 7 (D) 30

Show the solution
  1. Write n! = n × (n − 1) × (n − 2)!.
  2. The equation becomes n(n − 1) × (n − 2)! = 30 × (n − 2)!.
  3. Cancel (n − 2)!: n(n − 1) = 30.
  4. Two consecutive numbers with product 30 are 6 and 5, so n = 6.
  5. Algebraically: n² − n − 30 = 0, so (n − 6)(n + 5) = 0, giving n = 6 or n = −5.
  6. Reject n = −5 since a factorial needs a whole number.
  7. Check: 6! = 720 and 30 × 4! = 30 × 24 = 720.

Answer: n = 6 (option B)

Exam tips

  • Memorise factorials up to 7! so you can check options quickly.
  • Questions often hide a common factorial. Look for it first before multiplying anything.
  • If an option equals your answer only after using 0! = 0, you have fallen for a trap. Recheck.
  • For equations with n, reject negative roots and non-integers. This eliminates options fast.
  • With no negative marking, always attempt the question. Testing an option by substitution is usually quicker than solving fully.

Practice questions from Permutation and Combinations

Factorial Notation and Its Properties in other exams

The same ground in other exams, if you are preparing for more than one or want another angle on it.

Factorial Notation and Its Properties: frequently asked questions

Why is 0 factorial equal to 1?

It is defined that way so that n! = n × (n − 1)! holds for n = 1. Putting n = 1 gives 1! = 1 × 0!, so 0! = 1. It also matches the fact that there is one way to arrange nothing.

How do you simplify factorial expressions?

Expand the larger factorial until the smaller factorial appears, using n! = n × (n − 1)!. Cancel the common factorial and multiply the few terms left. This avoids large calculations.

Can you find the factorial of a negative number or a fraction?

Not at this level. In CMA Foundation, n! is defined only for zero and positive whole numbers. If a solved value of n is negative or fractional, reject it.

Is (m + n)! equal to m! + n!?

No. Factorial does not distribute over addition or multiplication. Compute the sum or product inside the bracket first, then take its factorial.