EQN-05-04

The Counting Principle

Count arrangements and choices by multiplying options stage by stage — the foundation of counting questions.

~5 min read•Difficulty 2/5

Multiply the choices

If one stage has m options and the next has n, together they have m × n outcomes. Extend across as many stages as needed.

3 shirts \times 4 trousers = 12 outfits

Arrangements (order matters)

Filling n distinct positions from n items: multiply the shrinking options.

Arrange 4 books on a shelf : 4 \times 3 \times 2 \times 1 = 24 ways (= 4!)

If you fill only some positions, stop early: choosing a 1st and 2nd from $5 \to 5 \times 4 = 20.$

Watch for restrictions

Handle a restricted slot first.

A 3 -digit code, first digit can’ t be 0, digits may repeat :

9 \times 10 \times 10 = 900 codes .

Order matters vs not

If swapping two chosen items gives the "same" selection (a committee, a handshake), order does not matter, and you must divide out the duplicate orderings.

Diagram

Counting principle: multiply the choices at each stage $(3 \times 4 \times 2) .$

Worked examples

Example 1 (difficulty 1/5)

A menu has 3 starters, 4 mains, and 2 desserts. How many different three-course meals are possible?

◎ Predict the answer first:

Example 2 (difficulty 2/5)

How many ways can a president and a vice-president be chosen from 6 people?

◎ Predict the answer first:

Tips

Multiply the number of options at each independent stage.
Fill the most restricted position first (e.g. "first digit can't be 0").
Different roles $\to$ order matters; identical membership $\to$ divide out repeats.

Common mistakes

Adding the options instead of multiplying

Why: "Total choices" sounds like a sum.

Fix: Independent successive choices multiply, not add.

Counting ordered selections when order should not matter

Why: The stage-by-stage method always produces ordered counts.

Fix: For committees/handshakes, divide by the number of orderings of the chosen group.

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