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Learn Index · Probability and Combinatorics Grade 9–10

Factorials

n! = n × (n − 1) × … × 1.

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What is a factorial?

n! (read "n factorial") means multiply all positive whole numbers from n down to 1:

n! = n × (n − 1) × (n − 2) × … × 2 × 1

n n!
1 1
2 2
3 6
4 24
5 120
6 720
7 5,040
10 3,628,800

Factorials grow incredibly fast. 20! is about 2.4 quintillion.

Zero factorial

0! = 1 by definition. This keeps formulas consistent: there is exactly one way to arrange zero objects (do nothing).

Why factorials matter: arrangements

The number of ways to arrange n different objects in a row is n!.

Example: How many ways can 4 books be arranged on a shelf?

  • 1st spot: 4 choices; 2nd: 3; 3rd: 2; 4th: 1
  • 4 × 3 × 2 × 1 = 4! = 24

Example: 6 runners finish a race. Possible finishing orders? 6! = 720.

These are permutations of all the items.

Simplifying factorial expressions

Write the larger factorial until it matches the smaller, then cancel.

  • 8!/6! = (8 × 7 × 6!)/6! = 8 × 7 = 56
  • 10!/(8! × 2!) = (10 × 9)/2 = 45
  • (n + 1)!/n! = n + 1

Never calculate huge factorials in full when you can cancel.

Arrangements with identical items

If some items are identical, divide by the factorial of each repeated group.

Example: Arrangements of the letters in BOOK:

4!/2! = 24/2 = 12 (the two O’s are identical)

Example: MISSISSIPPI (11 letters: 4 S, 4 I, 2 P):

11!/(4! × 4! × 2!) = 34,650

Arrangements in a circle

n people around a round table: (n − 1)!, because rotations are the same arrangement.

5 people → 4! = 24

Calculator

Most scientific calculators have an x! or n! key.

Worked examples

Example 1: 5! = 120.

Example 2: 12!/10! = 12 × 11 = 132.

Example 3: Arrange the letters of LEVEL: 5!/(2! × 2!) = 30.

Common mistakes

  • Thinking 0! = 0.
  • Cancelling incorrectly: 6!/3! is not 2!.
  • Forgetting identical items when counting arrangements.

Try it yourself

  1. 7!
  2. 9!/7!
  3. Arrange 5 different photos in a row.
  4. Arrange the letters of APPLE.
  5. Seat 7 people at a round table.

Answers: 1) 5,040 2) 72 3) 120 4) 60 5) 720

Quick recap

  • n! = n × (n − 1) × … × 1, and zero factorial 0! = 1.
  • n distinct items can be arranged in n! ways.
  • Simplify by cancelling before calculating.