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You have listed pairs in a table before. The lunch stall offers 3 breads, 5 fillings and 4 drinks. Every combination is available.
How many bread and filling pairs are there?
Tap to fill the grid, one at a time: 3 rows of 5.
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Counting Outcomes with the Product Rule
Count outcomes by multiplying stage counts when every partial outcome has the same number of next choices. Adjust for restrictions and count equally likely outcomes to find probabilities.
What a learner can do afterwards
- Multiply the options at each stage to count the total outcomes
- Adjust counts for no-repeat rules and other restrictions
- Find probabilities by counting wanted and total equally likely outcomes
1 · Read
You have listed pairs in a table. A complete list counts every allowed outcome once. Multiplication can count the same outcomes without writing them all. An outcome records one choice at each stage. Choosing no border is one option too, if it is allowed. If a required stage has 0 options, there are 0 complete outcomes.
Tap to fill the grid, one at a time: 3 rows of 5.
Each of 3 breads allows 5 fillings: 3 × 5 = 15 sandwiches. Each sandwich allows 4 drinks: 15 × 4 = 60 lunches. Multiply stage counts when every partial outcome has the same number of next choices. The options may differ in identity; their counts must match. Fixing a stage gives it 1 option. Adding a new one-option stage leaves the count unchanged. Adding 3 + 5 + 4 counts 12 menu options, not lunches.
Try it together
To find a missing stage count, divide the total by the product of the known counts. If every combination is allowed and 2 × ? × 7 = 56, the known counts give 2 × 7 = 14. The missing count is 56 / 14 = 4. Check: 2 × 4 × 7 = 56.
Give three different prizes, in a set order, to three different volunteers from a group of 8. There are 8, then 7, then 6 choices: 8 × 7 × 6 = 336. Swapping prize winners changes the outcome. For codes, count each position. Digits 0 to 9 give 10 choices. Repeats leave all 10 available; without repeats, each used digit is removed. A code may start with 0 unless forbidden. If 0 is forbidden first, there are 9 first choices; 0 is still available later.
Try it together
Raffle codes use one letter A to E, then two digits 0 to 9. Repeats and initial 0s are allowed. There are 5 × 10 × 10 = 500 codes. Each code is on one ticket, and all tickets are equally likely to be drawn. Fix A: 1 × 10 × 10 = 100 codes. Their probability is 100 / 500 = 0.2. Use wanted / total only for equally likely outcomes. Apply the same restrictions to both counts. If codes have different ticket counts, count equally likely tickets instead.
Tap to shade 1 of the 5 parts.
Each equal part represents 100 tickets. The shaded part represents the 100 tickets starting with A.
If the number of next choices varies, split the outcomes into cases that do not overlap. Count each case and add the counts. Suppose two cup sizes allow all 5 drinks, while a larger size allows only 3. The two sizes give 2 × 5 = 10 orders. Add the larger size's 3 orders: 10 + 3 = 13. Multiplying 3 × 5 would include 2 unavailable orders.
Multiply stage counts when each partial outcome has the same number of next choices. Adjust for restrictions. For equally likely outcomes, probability is wanted outcomes / total outcomes.
This lesson, question bank and flashcards were written for this dataset.
These readings, practice questions and flashcards were independently written for this dataset.
More to read
Count complete choices
A display uses one of 4 backgrounds and one of 6 headings. Every background works with every heading. Each background gives 6 displays, so there are 4 × 6 = 24 complete choices.
The sum 4 + 6 counts the individual options. It does not count the displays made by pairing them.
An optional stage needs care. If you can choose a frame or leave the display unframed, leaving it unframed is one allowed option. A required stage with no available option prevents any complete display.
Equal counts can use different options
Five walking routes each have exactly 3 possible starting points. The starting points differ between routes. There are still 5 × 3 = 15 route-and-start choices: every route contributes three.
Before multiplying, check the number of continuations after each partial outcome. The later options need not have the same names, but their counts must match at that stage.
A new stage with one option gives each existing outcome one continuation. Its factor is 1, so the count stays unchanged.
Track positions and used options
Two different jobs are filled by different people from a group of 7. The first job has 7 candidates. After that person is chosen, the second has 6. There are 7 × 6 = 42 assignments. Swapping the two job holders produces a different assignment.
For a code, positions also matter. A used digit remains available if repeats are allowed. Otherwise, remove it from the later choices.
Check the first-position rule separately. If a code cannot start with 0, its first position has only 9 digit choices. Zero can still be available in a later position.
Add separate case counts
A printer offers 4 paper types. Three allow 2 ink choices each. The fourth allows only 1 ink choice. A product of 4 × 2 would count an unavailable pairing.
The first three paper types give 3 × 2 = 6 prints. The remaining type gives 1. Add 6 + 1 = 7 allowed prints.
These cases do not overlap because a print uses exactly one paper type. You can multiply within a case with suitable stage counts, then add the separate case totals.
Count outcomes with equal chances
A ticket carries one of 7 symbols and one of 4 numbers. Every pairing is printed once, and every ticket has the same chance of being drawn. There are 7 × 4 = 28 tickets.
Fix one symbol. It still pairs with all 4 numbers, so 4 tickets have that symbol. Its probability is 4 / 28 = 1/7.
Use the same restrictions in the wanted and total counts. If some pairings appear on extra tickets, distinct pairings no longer have equal chances. Count the equally likely tickets instead. The count-based probability rule needs outcomes with equal chances.
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Take it off screen
Print a worksheetA4 with an answer key page for grown-ups. No screen, no internet.
Where it sits
Where this leads
Jobs that lean on this skill. Follow one to see everything it is built on.
Then practise
32 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.