Teach Me Maths

Product Rule for Counting

How to count choices and arrangements, handle restrictions, and avoid counting the same outcome twice.

Multiply the choices

When a task has successive stages, multiply the number of choices at each stage. If every first choice can be followed by each of the second choices, a first choices and b second choices give ab outcomes.

Count complete outcomes, not the total number of items on offer. If the number of choices changes depending on an earlier decision, split the problem into separate cases.

Worked example 1

A café offers 4 sandwich fillings, 3 breads and 2 drinks. A meal contains one of each. How many different meals are possible?

  1. For each filling there are 3 breads, giving 4\times3=12 sandwich choices.

  2. Your turn. Now include the drinks: work out 12\times2

    12\times2=24

  3. Answer: 24 meals. Adding 4+3+2 would count options, not complete meals.

Explore a table of outcomes

Try it: double one number of choices while holding the other fixed. The number of complete outcomes doubles.

Repetition changes the count

For a code that allows repetition, the same choices are available again at the next position. Without repetition, each used item is removed from later choices. Check whether the first digit may be 0: a code may start with zero, but a multi-digit number cannot.

Worked example 2

A code consists of two letters from A, B, C, D, E, followed by three digits from 0 to 9. Repetition is allowed. How many codes are possible?

  1. The five positions have 5, 5, 10, 10 and 10 choices. Zero is allowed in every digit position.

  2. Your turn. How many ordered pairs of letters are possible?

    5\times5=25

  3. Answer: 25\times10^3=25\,000 codes. AB and BA are different codes.

Worked example 3

Seven different books are available. How many ways can three of them be placed, in order, on a shelf?

  1. Choose the leftmost book in 7 ways. There are then 6 choices for the next book.

  2. Your turn. How many choices remain for the third book?

    There are 5 books left to choose from.

  3. Answer: 7\times6\times5=210 arrangements. The order matters, so do not divide by the number of orders.

Deal with the restricted position first

A final digit may need to be odd or even, or a particular person may need an end seat. Choose the most restricted position first, then count the remaining positions. The order in which you count positions need not be their written order.

Worked example 4

How many four-digit odd numbers can be made from 1,2,5,6,7,8, with no repeated digit?

  1. Your turn. How many choices are there for the final digit?

    The last digit is 1, 5 or 7: three choices.

  2. Once the final digit is fixed, fill the first three positions in 5\times4\times3 ways. There is no zero to exclude.
  3. Answer: 3\times5\times4\times3=180 numbers.

Add separate cases

Multiply choices within a case. Add totals for cases that cannot overlap and together cover every possibility. A useful split is whether the final digit is zero: that changes how many digits can go first.

Worked example 5

How many three-digit even numbers can be made from 0,2,3,5,6, with no repeated digit?

  1. If the last digit is 0, there are 4 choices for the first digit and 3 for the middle: 4\times3=12
  2. If the last digit is 2 or 6, there are 2 last-digit choices. The first digit has 3 non-zero choices; the middle then has 3 choices.

  3. Your turn. Work out 2\times3\times3

    There are 18 numbers with a non-zero final digit.

  4. Answer: 12+18=30 numbers. The cases are separate because a number has only one final digit.

Worked example 6

Five named runners enter a race with no ties. In how many finishing orders are two specified runners next to each other?

  1. Treat the two specified runners as one block. Together with the other three runners, this gives four distinct objects to arrange.

  2. Your turn. Work out 4\times3\times2\times1

    The four objects can be arranged in 24 ways.

  3. The pair can be ordered in 2 ways inside its block. Both must be counted.
  4. Answer: 24\times2=48 finishing orders.

Common mistakes

  • Adding successive choices. Multiply the number of choices at each stage.
  • Reusing a forbidden item. Reduce later choices when repetition is not allowed.
  • Allowing a leading zero in a number. Count the first digit separately; it cannot be zero.
  • Ignoring whether order matters. Decide whether rearranging the same items creates a different outcome.
  • Adding overlapping cases. Make cases mutually exclusive, or remove the double-counted outcomes.
  • Forgetting order inside a group. Count the arrangements within a grouped pair as well as its position.

Now try it: Product Rule for Counting practice questions

More on this topic: Product Rule for Counting worksheet with full solutions