All skills Math

Learn Index · Probability and Combinatorics Grade 9–10

Fundamental counting principle

If one choice has m options and another n, together there are m × n.

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Multiply the choices

The fundamental counting principle: if one choice can be made in m ways and a second in n ways, the two together can be made in m × n ways.

It extends to any number of stages: m × n × p × …

Example: outfits

3 shirts, 4 pants and 2 pairs of shoes:

3 × 4 × 2 = 24 outfits

Example: meals

A restaurant offers 4 appetizers, 6 mains and 3 desserts. Number of 3-course meals:

4 × 6 × 3 = 72

Repetition allowed vs not

With repetition (items can be reused):

  • 4-digit PIN using 0–9: 10 × 10 × 10 × 10 = 10,000

Without repetition (each item used once):

  • 4-digit PIN with no repeated digits: 10 × 9 × 8 × 7 = 5,040

Licence plates

Ontario passenger plates have 4 letters and 3 digits (ABCD 123):

26⁴ × 10³ = 456,976 × 1,000 = 456,976,000 possible plates (before removing restricted combinations).

Restrictions

Handle the restricted stage first.

Example: How many 3-digit numbers (100–999) are odd?

  • Hundreds digit: 1–9 → 9 choices
  • Tens digit: 0–9 → 10
  • Ones digit: 1, 3, 5, 7, 9 → 5
  • Total: 9 × 10 × 5 = 450

Example: How many 3-letter codes from A–Z start with a vowel and have no repeats?

  • 1st: 5 vowels; 2nd: 25; 3rd: 24 → 5 × 25 × 24 = 3,000

"Or" situations: add

If choices are separate cases (not stages), add.

Example: You can take the bus (3 routes) or the train (2 routes). Ways to travel: 3 + 2 = 5.

Connection to probability

P(guessing a 4-digit PIN in one try) = 1/10,000.

Worked examples

Example 1: Flip a coin, roll a die, spin a 4-colour spinner. Outcomes? 2 × 6 × 4 = 48.

Example 2: Passwords: 2 letters then 3 digits, repetition allowed. 26² × 10³ = 676,000.

Example 3: 5 runners. How many ways for 1st, 2nd, 3rd? 5 × 4 × 3 = 60.

Common mistakes

  • Adding when you should multiply (stages in sequence multiply).
  • Ignoring restrictions like "no leading zero".
  • Forgetting to reduce choices when repetition isn’t allowed.

Try it yourself

  1. 5 flavours of ice cream and 3 types of cone. Combinations?
  2. 3-digit codes from 0–9, repetition allowed?
  3. Same, no repetition?
  4. How many even 4-digit numbers (1000–9999)?
  5. 8 students; how many ways to choose a president and vice-president?

Answers: 1) 15 2) 1,000 3) 720 4) 4,500 5) 56

Quick recap

  • Fundamental counting principle: multiply the choices at each stage.
  • Watch for repetition and restrictions.
  • Add separate "or" cases.