The Counting Principle

What is the Counting Principle?

The Fundamental Counting Principle:

If one event can happen in m ways and another event can happen in n ways, the total number of ways both events can happen is:

m × n

This extends to any number of events: m × n × p × ...

Tree Diagrams

A tree diagram shows all possible outcomes visually.

Example: Coin and Die

Flip a coin, then roll a die.

Coin        Die         Outcome
Heads  ─┬─  1  →  H1
        ├─  2  →  H2
        ├─  3  →  H3
        ├─  4  →  H4
        ├─  5  →  H5
        └─  6  →  H6
Tails  ─┬─  1  →  T1
        ├─  2  →  T2
        ├─  3  →  T3
        ├─  4  →  T4
        ├─  5  →  T5
        └─  6  →  T6

Total outcomes: 2 × 6 = 12

Applying the Principle

Example 1: Outfits

3 shirts (red, blue, green) and 4 pants (black, grey, navy, white)

Total outfit combinations:

3 × 4 = 12 outfits

Example 2: Three Events

A restaurant offers:

• 3 starters
• 5 main courses
• 2 desserts

Total meal combinations:

3 × 5 × 2 = 30 different meals

Example 3: License Plates

A license plate has 3 letters followed by 3 digits (repetition allowed)

Total plates:

26 × 26 × 26 × 10 × 10 × 10
= 26³ × 10³
= 17,576 × 1,000
= 17,576,000 plates

With and Without Repetition

With Repetition

Each choice is independent — the same option can be used again.

Example: 4-digit PIN (digits 0–9, repetition allowed)

10 × 10 × 10 × 10 = 10⁴ = 10,000 PINs

Without Repetition

Once an option is used, it cannot be used again.

Example: First, second, and third place from 5 runners

5 × 4 × 3 = 60 ways

(5 choices for 1st, then 4 left for 2nd, then 3 left for 3rd)

Connecting to Permutations and Combinations

The Counting Principle is the foundation of:

Permutations — ordered arrangements (order matters)

Combinations — unordered selections (order doesn't matter)

Example: Choose 2 letters from {A, B, C}

With order (permutations): AB, BA, AC, CA, BC, CB → 6 ways = 3 × 2

Without order (combinations): AB, AC, BC → 3 ways

Practice