2.4 The product rule for counting — Learn

Prerequisites: multiplication and organised lists of possible outcomes.

Objectives: count combined choices by multiplying the number of options at each stage; use the rule when choices have restrictions.

Covers: N5 (the product rule for counting; Higher only).

Multiply choices across stages

The product rule says that if one choice has $a$ options and a following choice has $b$ options for each first choice, there are $a\times b$ combined outcomes.

Worked example. A meal has 3 main choices and 4 drink choices. How many different main-and-drink meals are possible?

Each of the 3 mains can be paired with 4 drinks.

So the total is $3\times4=12$ meals.

Worked example. A code has two different digits chosen from 1, 2, 3 and 4. How many codes are possible?

There are 4 choices for the first digit.

After choosing it, 3 digits remain for the second place.

So there are $4\times3=12$ codes.

The restriction changes the second number of choices, so it is not $4\times4$.

Watch out. Multiplying the same number at every stage can ignore restrictions. When digits cannot repeat, count the remaining options after the first choice.

Guided check: predict, test, explain

Predict: how many three-character codes can use A, B, C and D if letters may repeat?

Test: count the choices at each position and multiply them.

Explain: why does allowing repetition keep the number of choices the same at every position?

Progress check

The meal choices above can also be listed in a table. Explain how the table would contain 12 pairs and how multiplication counts the same pairs.

You should now be able to

• identify the stages in a counting problem;
• multiply the number of options at each stage;
• adjust a later stage when choices cannot repeat;
• explain why your multiplication matches the possible outcomes.