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This page sets out the method for systematic listing in the order you would use it, then walks through the full lesson explanation. For three complete worked examples with the slips that cost marks, see the Systematic listing topic page.
When you see 'list all the possible …', choose an order for your options and stick to it.
When you see 'how many ways' without needing the actual list, multiply the numbers of choices.
Fix the first choice, then list every second choice under it.
Then move to the next first choice.
If every option in the first list pairs with every option in the second, multiply.
2 tops and 3 sauces make 6 meals.
Red then blue is a different pair from blue then red only when order matters.
If the question says “a pair of colours” and order does not matter, those are one outcome.
Multiply a third choice on as well.
2 × 3 × 2 = 12 if each stage is independent.
The list should be as long as the product.
If it is shorter, something was skipped.
A grid with one choice along the side and one along the top shows every cell once.
Count the cells if the numbers are small.
Put the first group down the side and the second group along the top, either on paper or in your head. Maya has tea, juice and milk, and a bun or a biscuit. The tea row is tea with a bun, and tea with a biscuit. The juice row is the same two snacks. The milk row is the same two again. Six cells, six pairs.
Say a short list in that order. Tea-bun, tea-biscuit, juice-bun, juice-biscuit, milk-bun, milk-biscuit. Each drink appears twice, once for each snack. If a drink appears once, a snack was skipped.
An outfit is one top and one bottom. Writing red then jeans, and jeans then red, is the same outfit. Order matters for a number, because 416 and 461 are different. Decide whether order matters before you count.
Method: 1. Put one group down the side and the other along the top. 2. Fill every pairing, or count the cells instead of writing them all. 3. Each cell is one different choice.
For a three-digit number that uses each of three digits once, choose the hundreds digit first, then the tens, then the units. Digits 4, 1 and 6 give 3 choices for hundreds, 2 left for tens, and 1 for units. That is 3 × 2 × 1 = 6 numbers: 416, 461, 146, 164, 614, 641.
Write them in blocks by the first digit. Starting with 4: 416, 461. Starting with 1: 146, 164. Starting with 6: 614, 641. Each block has 2 × 1 = 2 numbers, and three blocks make 6.
A two-digit code from 1, 2 and 3, with no repeat, has three choices then two: 3 × 2 = 6. Allowing a repeat would be 3 × 3 = 9. Read whether each digit may be used once.
Method: 1. Fill the hundreds place, then the tens place, then the units place. 2. If each digit is used once, the choices fall by one each time. 3. Multiply those choices to get the number of arrangements.
An even number ends in 0, 2, 4, 6 or 8. Lock the units digit first, then arrange the rest. From digits 3, 6 and 8, used once, the even endings are 6 and 8. That is 2 choices for units, then 2 digits left, then 1: 2 × 2 × 1 = 4. The numbers are 386, 836, 368 and 638.
A number greater than 500 must start with 5 or more. From 3, 6 and 8, the hundreds digit is 6 or 8, not 3. Again 2 × 2 × 1 = 4: 638, 683, 836, 863. The numbers 368 and 386 are in the full list of 6, and they fail the filter.
Two filters need cases when they share a digit. From 4, 5 and 8, even and greater than 500, the first digit is 5 or 8 and the last digit is 4 or 8. Starting with 5 gives 584 and 548. Starting with 8 cannot also end in 8, so only 854. The list is 548, 584, 854, and 2 + 1 = 3.
Method: 1. Lock the digit that the rule fixes, such as an even units digit. 2. Arrange the remaining digits in the other places. 3. Count only the arrangements that pass the rule.
A "list all the possibilities" question is about being organised, not clever. The marks go to a list that is complete, has no repeats and is in a pattern you can follow. Fix the first item, write down every way to finish the list, then move to the next first item. For three letters A, B and C that gives ABC, ACB, then BAC, BCA, then CAB, CBA.
Real exam papers often print one answer for you with the words "one has been done for you". Use it. It tells you the format to copy, such as letters written together or pairs written in brackets. Count your finished list against what you expect. If the choices multiply to 6, a list of 5 or 7 items tells you straight away that something is missing or has been repeated.
Method: 1. Copy the format of the example the question gives you. 2. Fix the first item and list every possible ending before changing it. 3. Move to the next first item and repeat the same pattern. 4. Count your list and compare it with the number you expect from multiplying the choices.
Tip: If the question says "one has been done for you", cross it out of your plan so you do not write it twice.
Two details in the question change the list completely. The first is whether an item may be used twice. "The two scoops can be the same flavour or different flavours" allows vanilla and vanilla. The second is whether order matters. Two scoops of strawberry then mint is the same ice cream as mint then strawberry, so only one of the two should appear. A queue of songs is different: ABC is not the same as CAB.
Underline these words before you start. If order does not matter, list pairs in a fixed pattern, such as always writing the earlier letter first, and you will never list the same pair twice. If repeats are allowed, include the double items such as VV and SS, which students often forget because they look odd.
Method: 1. Underline "same", "different", "once each" and "in order" in the question. 2. Decide whether a repeat such as VV is allowed. 3. Decide whether reversing a pair makes a new item or the same item. 4. List in a fixed pattern and count the result.
Meal deals, outfits and lunch menus usually have three groups, such as a main, a side and a drink. You are often told to find only how many there are, not to list them. In that case multiply the number of choices in each group. A lunch with 5 sandwiches, 3 snacks and 4 drinks gives 5 × 3 × 4 = 60 different lunches.
The next step in an exam is a restriction: only vegetarian items, or one choice removed because it has sold out. Reduce the number of choices in the groups that are affected, then multiply again. "Show that there are 60 lunches" needs the full multiplication written out, because the answer is given. Never add the numbers of choices together. Adding counts the items on the menu, not the combinations.
Method: 1. Write the number of choices in each group next to its name. 2. Multiply them to count the combinations. 3. For a restriction, change the number of choices only in the affected groups. 4. Multiply again and write a sentence stating the answer.
Codes and number problems add rules to the counting. A padlock code with three digits, each chosen from 1, 2 and 3 with repeats allowed, has 3 choices for every place, so 3 × 3 × 3 = 27. If the question says "no digit is used twice", each place has one fewer choice than the one before: 3 × 2 × 1 = 6.
Rules about the last digit are the hardest. For an even number the last digit must be even, so fill that place first. Digits 2, 5, 7, 9 make a 3-digit even number only if the last digit is 2, which leaves 3 × 2 ways to fill the other two places. Fill the restricted place first, then the other places in order, and multiply.
Method: 1. Draw one box for each digit or place. 2. Fill the restricted place first and write how many choices it has. 3. Fill the other places, taking away used digits if no repeats are allowed. 4. Multiply the numbers in the boxes.
Tip: For "odd" or "even", look at the last digit first. For "greater than", look at the first digit first.
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