Systematic listing and the product rule

Counting outcomes is a steady earner on every WJEC paper — Foundation candidates list outcomes, Higher candidates apply the product rule and combinations.

Systematic listing

The trick is to fix one variable at a time and cycle through the others. To list every two-digit number formed from {2, 5, 7} with no repeats:

• First digit 2 → 25, 27
• First digit 5 → 52, 57
• First digit 7 → 72, 75

That's 6 numbers. Going systematically prevents missed cases — examiners deduct B-marks for omitted outcomes.

A second tool: a sample-space diagram (table). For two dice, draw a 6×6 grid showing all 36 ordered pairs.

The product rule for counting

If a task has k stages, with n_1 ways for stage 1, n_2 ways for stage 2, …, n_k ways for stage k, the total number of ways = n_1 × n_2 × … × n_k.

Example 1 — Menu choices

A canteen offers 3 starters, 5 mains, 4 desserts. Number of meals = 3 × 5 × 4 = 60.

Example 2 — Restricted PIN codes

A 4-digit PIN with each digit from 0–9 with repeats allowed: 10 × 10 × 10 × 10 = 10 000. With no repeats: 10 × 9 × 8 × 7 = 5040.

Example 3 — Arranging objects

Number of ways to arrange 5 different books on a shelf = 5! = 5 × 4 × 3 × 2 × 1 = 120.

Choosing with restrictions

WJEC Higher loves: "How many 4-digit numbers can you make from {1, 2, 3, 4, 5} with no repeats AND first digit even?"

• Even first digit: 2 choices (2 or 4).
• Remaining three positions: 4 × 3 × 2 = 24.
• Total: 2 × 24 = 48.

WJEC exam tip

For listing problems, write the strategy out loud — e.g. "fix the first digit, then list". This earns the M1 even if your final list is incomplete. For product-rule problems, write the multiplication line before evaluating; that single line is the M1.