--- title: "Counting Outcomes with the Product Rule" description: "Count the outcomes of several choices made in a row by multiplying the number of options at each stage, instead of listing them all." canonical: https://lightmysky.com/learn/mathematics/counting-outcomes-with-the-product-rule-mt_74gSbYIG8p source: https://lightmysky.com/learn/mathematics/counting-outcomes-with-the-product-rule-mt_74gSbYIG8p.md retrieved: 2026-09-12 --- > **Agent view.** This is the Markdown twin of the page, for tools and assistants. > When to use this site, and the call that answers each job: https://lightmysky.com/agent-instructions.md > API description (OpenAPI 3.1): https://lightmysky.com/openapi.json · Authentication: https://lightmysky.com/auth.md > Pricing: https://lightmysky.com/pricing.md · Catalog: https://lightmysky.com/llms.txt · Full catalog: https://lightmysky.com/llms-full.txt > Every machine-readable file on this domain: https://lightmysky.com/.well-known/ai-catalog.json > Ask for Markdown with `Accept: text/markdown`, a `.md` address, or `?mode=agent`. # Counting Outcomes with the Product Rule Count the outcomes of several choices made in a row by multiplying the number of options at each stage, instead of listing them all. Subject: Mathematics · Area: Probability · Ages 15 to 16 Page: https://lightmysky.com/learn/mathematics/counting-outcomes-with-the-product-rule-mt_74gSbYIG8p ## Ready when they can - Multiply the options at each stage to count the total outcomes - Adjust the count when an option cannot be used twice - Use the count as the denominator of a probability ## Lesson: Counting Without Listing Last stop you got a probability by throwing hoops, because nothing about a hoop could be reasoned out. Plenty of chances at this fair can be reasoned out, and to do it you need to know how many outcomes there are. Listing them works until it does not. The food stall is doing a lunch deal: pick one of 3 breads, one of 5 fillings and one of 4 drinks. Writing out every possible lunch would take a page, and the answer is one line of arithmetic. *(drawing: Every bread paired with every filling: 3 rows of 5, which is 15 sandwiches before anybody has chosen a drink.)* The grid shows the rule. Each of the 3 breads can take any of the 5 fillings, so there are 3 × 5 = 15 sandwiches. Now each of those 15 sandwiches can take any of the 4 drinks, so there are 15 × 4 = 60 lunches. That is the product rule: when a choice is made in stages, multiply the number of options at each stage. It works because each option at one stage opens up the whole set at the next. It runs backwards too: divide the total by the stages you know to find a missing one. *(drawing: One line instead of a page. Adding would give 12, which is not a count of anything here.)* Sometimes an option cannot be used twice. Three prizes are going to three different volunteers out of the 8 running the stalls. The first prize can go to any of the 8. Once it has gone, only 7 people are left for the second, and 6 for the third. So the count is 8 × 7 × 6 = 336. Each stage has one option fewer than the last, because whoever was picked is now out. **Example.** The count you work out is often the bottom of a probability. The raffle prints tickets as a letter from A to E followed by two digits, so there are 5 × 10 × 10 = 500 different codes, one ticket each. How many start with A? The letter is fixed and the digits are free: 1 × 10 × 10 = 100. So the chance of drawing a ticket starting with A is 100 / 500 = 0.2. Two things have to be true before you multiply. The stages must be separate choices, so that every combination really is on offer. If a drink comes with only one bread, count that drink on its own and add it to the rest. And for the count to be a denominator, the outcomes have to be equally likely. The 500 codes divide the chance evenly only because each is printed on one ticket. **Recap.** When a choice is made in stages, the number of outcomes is the options at each stage multiplied together. If something cannot be picked twice, each stage has one option fewer than the one before it. The count is often the denominator of a probability: count the outcomes you want, count all of them, and divide. Multiplying is only right when the stages are separate choices and every combination is genuinely available. ## Practice 32 questions on this page, each with its working shown. ## Needs first - [Venn Diagrams and Counting Outcomes](https://lightmysky.com/learn/mathematics/venn-diagrams-and-counting-outcomes-mt_eSv_w46u6H) - [Systematic Listing](https://lightmysky.com/learn/mathematics/systematic-listing-mt_gu0NPDvlhY) ## Opens up - [Counting with the Product and Sum Rules](https://lightmysky.com/learn/mathematics/counting-with-the-product-and-sum-rules-mt_3NZC5F7RKP) - [The Comparison-Sorting Lower Bound](https://lightmysky.com/learn/computing/the-comparison-sorting-lower-bound-mt_7AfUsPV-jW) - [The Addition Rule and Mutually Exclusive Events](https://lightmysky.com/learn/mathematics/the-addition-rule-and-mutually-exclusive-events-mt_QsIusFefeE) - [The Binomial Distribution](https://lightmysky.com/learn/mathematics/the-binomial-distribution-mt_X2kVGK67Md) - [The Binomial Expansion](https://lightmysky.com/learn/mathematics/the-binomial-expansion-mt_z16HZkNZ8c)