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.

W_1, W_2, W_3, W_4, K_1, K_2, K_3

Counting principle worksheet for Algebra 2

The fundamental counting principle: if one thing can happen in m ways and the next in n ways, together they happen in m times n ways. That one sentence is the whole topic, and it is what permutations and combinations are built on.

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

Select your difficulty

Pick more than one for a sheet that mixes them.

The preview can look cramped on a phone. The PDF and printed sheet come out normal.



Sigma Prep · sigmaprep.io/worksheets Name The Counting Principle Date Period Answer each question. Count the outcomes How many outcomes are there from a marble drawn from a bag of 7, then a letter drawn from A to E, then a spinner with 5 equal parts? An ice cream shop has 4 flavors and 3 toppings. How many ways are there to choose one of each? How many outcomes are there from a twelve-sided die, then a spinner with 6 equal parts? How many outcomes are there from a spinner with 5 equal parts, then a spinner with 6 equal parts? A cafe offers 7 sandwiches, 9 soups and 4 drinks. How many ways are there to choose one of each? How many outcomes are there from an eight-sided die, then a marble drawn from a bag of 7, then a spinner with 6 equal parts? A new car comes in 9 colors, 7 interior trims and 5 wheel styles. How many ways are there to choose one of each? A closet holds 2 shirts and 4 pairs of pants. How many ways are there to choose one of each? A pizza shop offers 7 crusts and 2 sauces. How many ways are there to choose one of each? How many outcomes are there from a four-sided die, then a ten-sided die, then a coin? Count codes, PINs and plates How many 5-letter codes can be made from the 26 letters if no letter repeats? A license plate has 2 letters followed by 4 digits. How many plates are possible if letters and digits may repeat? How many 5-digit PINs are possible if no digit repeats? A license plate has 1 letter followed by 3 digits. How many plates are possible if letters and digits may repeat? A lock has 3 dials, each numbered 1 to 7. How many settings are possible? A license plate has 2 letters followed by 1 digit. How many plates are possible if letters and digits may repeat? A license plate has 1 letter followed by 4 digits. How many plates are possible if letters and digits may repeat? A license plate has 1 letter followed by 1 digit. How many plates are possible if letters and digits may repeat? A license plate has 3 letters followed by 3 digits. How many plates are possible if letters and digits may repeat? A lock has 4 dials, each numbered 1 to 20. How many settings are possible? Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

The Counting Principle

Date Period

Answer each question.

Count the outcomes
How many outcomes are there from a marble drawn from a bag of 7, then a letter drawn from A to E, then a spinner with 5 equal parts?
An ice cream shop has 4 flavors and 3 toppings. How many ways are there to choose one of each?
How many outcomes are there from a twelve-sided die, then a spinner with 6 equal parts?
How many outcomes are there from a spinner with 5 equal parts, then a spinner with 6 equal parts?
A cafe offers 7 sandwiches, 9 soups and 4 drinks. How many ways are there to choose one of each?
How many outcomes are there from an eight-sided die, then a marble drawn from a bag of 7, then a spinner with 6 equal parts?
A new car comes in 9 colors, 7 interior trims and 5 wheel styles. How many ways are there to choose one of each?
A closet holds 2 shirts and 4 pairs of pants. How many ways are there to choose one of each?
A pizza shop offers 7 crusts and 2 sauces. How many ways are there to choose one of each?
How many outcomes are there from a four-sided die, then a ten-sided die, then a coin?

Count codes, PINs and plates
How many 5-letter codes can be made from the 26 letters if no letter repeats?
A license plate has 2 letters followed by 4 digits. How many plates are possible if letters and digits may repeat?
How many 5-digit PINs are possible if no digit repeats?
A license plate has 1 letter followed by 3 digits. How many plates are possible if letters and digits may repeat?
A lock has 3 dials, each numbered 1 to 7. How many settings are possible?
A license plate has 2 letters followed by 1 digit. How many plates are possible if letters and digits may repeat?
A license plate has 1 letter followed by 4 digits. How many plates are possible if letters and digits may repeat?
A license plate has 1 letter followed by 1 digit. How many plates are possible if letters and digits may repeat?
A license plate has 3 letters followed by 3 digits. How many plates are possible if letters and digits may repeat?
A lock has 4 dials, each numbered 1 to 20. How many settings are possible?



Sigma Prep · sigmaprep.io/worksheets Name The Counting PrincipleAnswer keyVersion 1 Date Period Answer each question. Count the outcomes How many outcomes are there from a marble drawn from a bag of 7, then a letter drawn from A to E, then a spinner with 5 equal parts? $175$ An ice cream shop has 4 flavors and 3 toppings. How many ways are there to choose one of each? $12$ How many outcomes are there from a twelve-sided die, then a spinner with 6 equal parts? $72$ How many outcomes are there from a spinner with 5 equal parts, then a spinner with 6 equal parts? $30$ A cafe offers 7 sandwiches, 9 soups and 4 drinks. How many ways are there to choose one of each? $252$ How many outcomes are there from an eight-sided die, then a marble drawn from a bag of 7, then a spinner with 6 equal parts? $336$ A new car comes in 9 colors, 7 interior trims and 5 wheel styles. How many ways are there to choose one of each? $315$ A closet holds 2 shirts and 4 pairs of pants. How many ways are there to choose one of each? $8$ A pizza shop offers 7 crusts and 2 sauces. How many ways are there to choose one of each? $14$ How many outcomes are there from a four-sided die, then a ten-sided die, then a coin? $80$ Count codes, PINs and plates How many 5-letter codes can be made from the 26 letters if no letter repeats? $7,893,600$ A license plate has 2 letters followed by 4 digits. How many plates are possible if letters and digits may repeat? $6,760,000$ How many 5-digit PINs are possible if no digit repeats? $30,240$ A license plate has 1 letter followed by 3 digits. How many plates are possible if letters and digits may repeat? $26,000$ A lock has 3 dials, each numbered 1 to 7. How many settings are possible? $343$ A license plate has 2 letters followed by 1 digit. How many plates are possible if letters and digits may repeat? $6760$ A license plate has 1 letter followed by 4 digits. How many plates are possible if letters and digits may repeat? $260,000$ A license plate has 1 letter followed by 1 digit. How many plates are possible if letters and digits may repeat? $260$ A license plate has 3 letters followed by 3 digits. How many plates are possible if letters and digits may repeat? $17,576,000$ A lock has 4 dials, each numbered 1 to 20. How many settings are possible? $160,000$ Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

The Counting PrincipleAnswer keyVersion 1

Date Period

Answer each question.

Count the outcomes
How many outcomes are there from a marble drawn from a bag of 7, then a letter drawn from A to E, then a spinner with 5 equal parts? $175$
An ice cream shop has 4 flavors and 3 toppings. How many ways are there to choose one of each? $12$
How many outcomes are there from a twelve-sided die, then a spinner with 6 equal parts? $72$
How many outcomes are there from a spinner with 5 equal parts, then a spinner with 6 equal parts? $30$
A cafe offers 7 sandwiches, 9 soups and 4 drinks. How many ways are there to choose one of each? $252$
How many outcomes are there from an eight-sided die, then a marble drawn from a bag of 7, then a spinner with 6 equal parts? $336$
A new car comes in 9 colors, 7 interior trims and 5 wheel styles. How many ways are there to choose one of each? $315$
A closet holds 2 shirts and 4 pairs of pants. How many ways are there to choose one of each? $8$
A pizza shop offers 7 crusts and 2 sauces. How many ways are there to choose one of each? $14$
How many outcomes are there from a four-sided die, then a ten-sided die, then a coin? $80$

Count codes, PINs and plates
How many 5-letter codes can be made from the 26 letters if no letter repeats? $7,893,600$
A license plate has 2 letters followed by 4 digits. How many plates are possible if letters and digits may repeat? $6,760,000$
How many 5-digit PINs are possible if no digit repeats? $30,240$
A license plate has 1 letter followed by 3 digits. How many plates are possible if letters and digits may repeat? $26,000$
A lock has 3 dials, each numbered 1 to 7. How many settings are possible? $343$
A license plate has 2 letters followed by 1 digit. How many plates are possible if letters and digits may repeat? $6760$
A license plate has 1 letter followed by 4 digits. How many plates are possible if letters and digits may repeat? $260,000$
A license plate has 1 letter followed by 1 digit. How many plates are possible if letters and digits may repeat? $260$
A license plate has 3 letters followed by 3 digits. How many plates are possible if letters and digits may repeat? $17,576,000$
A lock has 4 dials, each numbered 1 to 20. How many settings are possible? $160,000$

How to do these

How many outcomes are there from a coin, then a six-sided die?

The coin has 2 outcomes and the die has 6.
Multiply them.

The answer is $12$ .

Where students go wrong

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.

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. → ${W_{1},$ $W_{2},$ $W_{3},$ $W_{4},$ $K_{1},$ $K_{2},$ $K_{3}}$

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.

Algebra 2 is usually taken around 10th or 11th 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.

Counting principle worksheet for Algebra 2

Select your difficulty

Count the outcomes

Count codes, PINs and plates

Count the outcomes

Count codes, PINs and plates

How to do these

Where students go wrong

What you can put on this worksheet

Questions about these worksheets

Are these counting principle worksheets free?+

Do the counting principle worksheets come with an answer key?+

Are these counting principle worksheets printable?+

What grade are these counting principle worksheets for?+

Can I get a different version of the same counting principle worksheet?+

Is this a free alternative to Kuta Software for counting principle?+

Can I choose how hard the counting principle questions are?+

What kinds of question are on a counting principle worksheet?+

What do students get wrong with counting principle?+