Systematic listing and the product rule

OCR sets at least one counting question per Foundation series. The marks come from showing a systematic approach (not just guessing combinations).

Systematic listing

To list outcomes systematically, fix one variable at a time and vary the others.

Example: How many 2-digit numbers can be made using digits 1, 2, 3 without repetition?

• Tens = 1: units 2, 3 → 12, 13.
• Tens = 2: units 1, 3 → 21, 23.
• Tens = 3: units 1, 2 → 31, 32.
• Total: 6.

The approach: fix the tens, list each possible units. Repeat.

The product rule (multiplication principle)

If choice A has m options and choice B has n options (independent), the total combined choices = m × n.

Example: a sandwich shop has 4 breads × 3 fillings × 2 cheeses = 24 sandwich combinations.

For ordered selections without repetition (permutations):

• 1st pick: n options.
• 2nd pick: n − 1 options.
• 3rd pick: n − 2 options.
• Total = n(n−1)(n−2)…

Example: how many 3-letter codes from A, B, C, D without repetition?

• 4 × 3 × 2 = 24.

With repetition vs without

• WITH repetition: each slot has full choice. E.g. 4-digit PIN with digits 0–9 = 10⁴ = 10 000.
• WITHOUT repetition: each slot has one fewer option than the last.

Combinations vs permutations

• Permutation: order matters (codes, queues).
• Combination: order doesn't matter (committees, hands of cards).

GCSE rarely requires the formal nCr formula but expects students to know to divide by repetitions when order doesn't matter.

Example: How many ways to choose 2 students from 4 to form a (non-ordered) committee?

• Permutations: 4 × 3 = 12.
• But pairs AB and BA are the same → divide by 2.
• Combinations: 12/2 = 6.

OCR mark scheme conventions

• M1 for stating the product rule explicitly (or systematic listing).
• A1 for the correct count.
• "List all possibilities" → expects systematic approach, not random.