When a question asks for every possible outcome, work in a fixed order so that nothing is missed or repeated. List combinations by pairing the first item of one set with every item of the other, then moving on. Once the pattern is clear, count without listing: multiply the number of choices at each stage.
Counting outcomes
- List in a fixed order: fix the first choice, run through the rest, then move on
- Combinations of two sets: number in set 1 × number in set 2
- Several stages: multiply the number of choices at each stage
- Arrangements of n different items in a line: n × (n − 1) × … × 1
Listing in a systematic order
The key to listing is a fixed order. Choose the first item, list everything that can follow it, then move to the next first item. For two-digit numbers from 4, 5 and 6 with no repeated digit, list 45, 46, then 54, 56, then 64, 65. Working this way, you can see that every possibility has been found.
Listing combinations from two sets
To list every combination of one item from each of two sets, take the first item from the first set and pair it with every item from the second, then do the same for the next item. A table, with one set across the top and the other down the side, shows the same combinations in a grid.
Counting by multiplying
You do not always need to list every case. If there are 3 choices of starter and 5 choices of main course, every starter can go with every main, giving 3 × 5 = 15 meals. Add a third stage with 2 desserts and there are 3 × 5 × 2 = 30 meals. This is called the product rule for counting.
When repeats are or are not allowed
Read carefully whether items can be repeated. With repeats allowed, a 3-digit code from the digits 0 to 9 has 10 × 10 × 10 = 1000 possibilities. Without repeats, the number of choices drops by one at each stage, so arranging 5 different books on a shelf gives 5 × 4 × 3 × 2 × 1 = 120 orders.
Answering follow-up questions
Exam questions often ask how many of the listed outcomes satisfy a condition, such as being even or greater than 50. Use your complete list and count the ones that fit; a systematic list makes this reliable.
Worked examples
Example 1. List all the two-digit numbers that can be made from the digits 4, 5 and 6, with no digit repeated.
- 4 first: 45, 46.
- 5 first: 54, 56.
- 6 first: 64, 65.
Answer: 45, 46, 54, 56, 64, 65.
Example 2. A café offers 2 sandwiches (cheese, ham) and 3 drinks (tea, coffee, juice). List every sandwich-and-drink combination.
- Cheese with each drink: cheese–tea, cheese–coffee, cheese–juice.
- Ham with each drink: ham–tea, ham–coffee, ham–juice.
Answer: 6 combinations, which matches 2 × 3.
Example 3. Using the digits 3, 4 and 7 once each, how many two-digit even numbers can be made?
- An even number must end in 4.
- The first digit can then be 3 or 7: 34 and 74.
Answer: Two numbers: 34 and 74.
Example 4. A restaurant has 3 starters, 5 mains and 2 desserts. How many different three-course meals are possible?
- Every combination of choices is possible, so multiply.
- 3 × 5 × 2.
Answer: 30 meals.
Example 5. A padlock code uses three digits, each from 0 to 9, and digits may repeat. How many codes are possible?
- Each digit has 10 choices.
- 10 × 10 × 10.
Answer: 1000 codes.
Common mistakes
- Listing at random. Use a fixed order so that no outcome is missed or repeated.
- Adding instead of multiplying. 3 starters and 5 mains give 3 × 5 meals, not 3 + 5.
- Ignoring “no repeats”. Without repeats, the number of choices falls by one at each stage.
- Counting the same pair twice. For combinations of two sets, pair each item of the first set once with each item of the second.
- Not answering the follow-up. After listing, count the outcomes that meet the stated condition.
How systematic listing is examined
Listing outcomes is a Foundation topic for GCSE and International GCSE students; the product rule for counting also appears on Higher papers. Questions ask you to:
- list all possible numbers, codes or combinations
- state how many outcomes satisfy a condition
- count combinations of meals, outfits or routes by multiplying
- work out the number of possible codes, with or without repeats
There is a systematic listing worksheet with answers, and the GCSE number revision engine gives unlimited practice.
Counting outcomes leads into the product rule for counting at Higher tier and sample space diagrams in probability.
Frequently asked questions
What does it mean to list systematically?
To list outcomes in a fixed, logical order, so you can be sure none are missing or repeated.
How do you work out the number of combinations?
Multiply the number of choices at each stage. Two shirts and three hats give 2 × 3 = 6 outfits.
Does the order of listing matter?
Not for the answer, but using a consistent order is what stops you missing outcomes.