--- title: "Counting Outcomes with the Product Rule" description: "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." 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: 2026-11-12 --- > **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 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. 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 counts for no-repeat rules and other restrictions - Find probabilities by counting wanted and total equally likely outcomes ## Lesson: Counting without listing 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. *(drawing: A stall offers 3 breads, 5 fillings and 4 drinks. Every combination is available. Each row represents a bread; each dot is a bread and filling pair.)* 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. **Example.** 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. **Example.** 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. 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. **Recap.** 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.* ## 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)