Worksheets · Geometry

Counting principle worksheet for Geometry

Geometry uses counting to set up probability: before you can say how likely an outcome is, you need the size of the sample space. Menus, license plates, locks and codes are counted by multiplying the options at each stage, with fewer options left when nothing can repeat.

Also taught in Algebra 2, Precalculus and Statistics, each at its own level and with its own examples.

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Pick more than one for a sheet that mixes them.

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Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

The Counting Principle

Date Period

Answer each question.

  1. Count the outcomes

    A cafe offers 8 sandwiches, 6 soups and 6 drinks. How many ways are there to choose one of each?
  2. A pizza shop offers 6 crusts, 9 sauces and 9 cheeses. How many ways are there to choose one of each?
  3. How many outcomes are there from a coin, then a letter drawn from A to E, then a spinner with 8 equal parts?
  4. How many outcomes are there from a coin, then a six-sided die?
  5. How many outcomes are there from a coin, then a ten-sided die, then a twelve-sided die?
  6. How many outcomes are there from a card drawn from a standard 52-card deck, then a marble drawn from a bag of 7?
  7. A new car comes in 6 colors and 5 interior trims. How many ways are there to choose one of each?
  8. An ice cream shop has 5 flavors and 4 toppings. How many ways are there to choose one of each?
  9. How many outcomes are there from a ten-sided die, then a card drawn from a standard 52-card deck, then a marble drawn from a bag of 7?
  10. How many outcomes are there from a spinner with 8 equal parts, then a twelve-sided die, then an eight-sided die?

  1. Count codes, PINs and plates

    How many 4-digit PINs are possible if no digit repeats?
  2. A license plate has 1 letter followed by 3 digits. How many plates are possible if letters and digits may repeat?
  3. A lock has 3 dials, each numbered 1 to 12. How many settings are possible?
  4. A license plate has 2 letters followed by 3 digits. How many plates are possible if letters and digits may repeat?
  5. How many 6-letter codes can be made from the 26 letters if no letter repeats?
  6. A license plate has 1 letter followed by 1 digit. How many plates are possible if letters and digits may repeat?
  7. A lock has 4 dials, each numbered 1 to 7. How many settings are possible?
  8. A license plate has 3 letters followed by 1 digit. How many plates are possible if letters and digits may repeat?
  9. A license plate has 2 letters followed by 1 digit. How many plates are possible if letters and digits may repeat?
  10. How many 3-letter codes can be made from the 26 letters if no letter repeats?

Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

The Counting PrincipleAnswer keyVersion 3

Date Period

Answer each question.

  1. Count the outcomes

    A cafe offers 8 sandwiches, 6 soups and 6 drinks. How many ways are there to choose one of each?288
  2. A pizza shop offers 6 crusts, 9 sauces and 9 cheeses. How many ways are there to choose one of each?486
  3. How many outcomes are there from a coin, then a letter drawn from A to E, then a spinner with 8 equal parts?80
  4. How many outcomes are there from a coin, then a six-sided die?12
  5. How many outcomes are there from a coin, then a ten-sided die, then a twelve-sided die?240
  6. How many outcomes are there from a card drawn from a standard 52-card deck, then a marble drawn from a bag of 7?364
  7. A new car comes in 6 colors and 5 interior trims. How many ways are there to choose one of each?30
  8. An ice cream shop has 5 flavors and 4 toppings. How many ways are there to choose one of each?20
  9. How many outcomes are there from a ten-sided die, then a card drawn from a standard 52-card deck, then a marble drawn from a bag of 7?3640
  10. How many outcomes are there from a spinner with 8 equal parts, then a twelve-sided die, then an eight-sided die?768

  1. Count codes, PINs and plates

    How many 4-digit PINs are possible if no digit repeats?5040
  2. A license plate has 1 letter followed by 3 digits. How many plates are possible if letters and digits may repeat?26,000
  3. A lock has 3 dials, each numbered 1 to 12. How many settings are possible?1728
  4. A license plate has 2 letters followed by 3 digits. How many plates are possible if letters and digits may repeat?676,000
  5. How many 6-letter codes can be made from the 26 letters if no letter repeats?165,765,600
  6. A license plate has 1 letter followed by 1 digit. How many plates are possible if letters and digits may repeat?260
  7. A lock has 4 dials, each numbered 1 to 7. How many settings are possible?2401
  8. A license plate has 3 letters followed by 1 digit. How many plates are possible if letters and digits may repeat?175,760
  9. A license plate has 2 letters followed by 1 digit. How many plates are possible if letters and digits may repeat?6760
  10. How many 3-letter codes can be made from the 26 letters if no letter repeats?15,600

Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

How to do these

How many 3-digit PINs are possible if no digit repeats?

  1. First digit: 10 choices.
  2. Second digit: 9 left, since one is used. Third digit: 8.
  3. 10 × 9 × 8.

The answer is 720.

Where students go wrong

Using the full set at every stage when items cannot repeat. A 4-digit PIN with no repeated digit is 10 × 9 × 8 × 7 = 5,040, not 10⁴ = 10,000, because each digit used is no longer available.

Also called fundamental counting principle, sample spaces or multiplication principle.

What you can put on this worksheet

How many outcomes altogether
How many outcomes are there from a six-sided die, then a card drawn from a standard 52-card deck? → 312
List the sample space
List the sample space for picking a marble that is purple or white and flipping a coin. → {PH, PT, WH, WT}
Codes, PINs and plates
A license plate has 3 letters followed by 1 digit. How many plates are possible if letters and digits may repeat? → 175,760
Arrange all of them: n!
4 runners finish a race with no ties. In how many different orders can they finish? → 24
Sample space when the second pick must be different
A team is Ada, Eli and Omar. One is picked as president, then a different one as vice president. List the sample space, using first letters (AE means Ada then Eli). → {AE, AO, EA, EO, OA, OE}
Identical objects, told apart by subscripts
A bag has 4 tiles marked W and 3 marked K. One is drawn. List the sample space, using subscripts to tell tiles with the same letter apart. → {W1, W2, W3, W4, K1, K2, K3}

Questions about these worksheets

Yes. Every counting principle sheet is free to print and free to download as a PDF. There is no account to make, no email to hand over and no limit on how many you take.

Yes. The answer key prints on a second page, and you can choose whether it shows each question beside its answer or just the answers in a list.

Yes. Print straight from the page, or download the PDF and print that. The sheet is laid out for paper rather than squeezed off a screen, so there is room to work under each question, and the answer key comes out on its own page.

Geometry is usually taken around 9th or 10th grade, though schools vary and plenty of students meet it earlier or later. Pick the difficulty rather than the grade: the easy level suits a first lesson on it, the hard level suits review before a test.

Yes, as many as you like. The questions are built fresh each time rather than picked from a fixed set of files, so pressing Generate gives a new sheet. That is what you want for a second class, a retake, or two students sitting next to each other.

That is the closest comparison, and Kuta Software is good software. It is also paid software you install on a computer. These worksheets run in a browser tab for free, with no account and nothing to install. Like Kuta, the counting principle questions are generated when you ask for them rather than pulled from a fixed set of files, so two classes never get the same sheet, and the answer key comes with it.

Yes. There are three levels, and you can tick more than one for a sheet that mixes them, in which case the questions come out easiest first. The harder levels are not just larger numbers: they ask for something the easy ones do not.

You pick from 6: how many outcomes altogether, list the sample space, codes, pins and plates, arrange all of them: n!, sample space when the second pick must be different and identical objects, told apart by subscripts. Tick as many as you want and set how many of each, or let it spread them evenly.

Adding instead of multiplying. Two outcomes then six outcomes is twelve, not eight, because every one of the two pairs with every one of the six.