Systematic Listing
List all possible pairs made by choosing one option from each of two groups.
What a learner can do afterwards
- List all possible meal combinations from 3 starters and 4 mains
- Organise pairs in a table so each possible pair appears once
- Explain why the total equals the product of the two group sizes when every pairing is allowed
1 · Read
You have used multiplication to find totals. Here, a combination means one choice from each of two groups. Two choices from the same group do not make a complete pair. When every pairing is allowed, multiply the two group sizes to count the different pairs. Adding the group sizes counts the separate options.
A cafe offers 4 starters and 5 mains. Any starter can go with any main. Pair the first starter with all 5 mains. Repeat for each remaining starter. There are 4 rows of 5 pairs: 4 x 5 = 20 meals. Follow the same main order for every starter.
Label rows with hats and columns with scarves. In each cell, combine its row's hat and its column's scarf:
| Hat | red scarf | blue scarf |
|---|---|---|
| cap | cap-red | cap-blue |
| hood | hood-red | hood-blue |
For the hood row and blue scarf column, write hood-blue. Fill every cell once to include every pair. Finish one row before starting the next. Either group can go down the side. One fixed hat has 2 scarf pairs.
Check the pairs as well as the entry count. With these hats and scarves, four entries could still repeat cap-red and leave out hood-blue. Compare each expected pair with the list. If all listed pairs are valid and different, subtract their count from the expected total to find how many are missing.
If a pair is unavailable, leave it out. For 4 ticket types and 6 days, there would be 4 x 6 = 24 pairs. If exactly one pair is unavailable, there are 24 - 1 = 23 available pairs. Adding one new day adds 4 pairs, one for each ticket type.
You can also work backwards from a total. Suppose every pattern comes in 4 colours and there are 12 different pairs. Group the pairs into rows of 4: 4, 8, 12. That makes 3 rows, so there are 3 patterns. This works because every pattern has all 4 colour choices.
Pair each option with every allowed option in the other group. Check that each pair appears once. When every pairing is allowed, multiply the two group sizes to check the total.
Lesson, questions and flashcards written for this dataset.
Additional questions and reading written for this dataset.
More to read
Checking a complete list
A museum offers two map styles, pocket and folded. Either map can carry a round, square or triangle stamp. A visitor chooses one map style and one stamp.
Start with pocket maps. List pocket-round, pocket-square and pocket-triangle. Then list folded-round, folded-square and folded-triangle. Two rows of three give six different pairs.
You could list stamp by stamp instead. Round would pair with pocket and folded. Square would pair with both, then triangle would pair with both. The order changes, but the six pairs stay the same.
To check a list, compare it with each expected pair. Counting six entries is not enough. A list could contain pocket-round twice and leave out folded-triangle. It would still have six entries, but one pair would be missing.
Give each map style a table row and each stamp a column. Put the matching map-stamp pair in every cell once. Then check that your list contains each table pair exactly once.
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