Worksheets · Precalculus

Permutations and combinations worksheet for Precalculus

Precalculus counting comes as word problems: finishing orders and offices where order matters, groups where it does not, and arrangements of words with repeated letters. It feeds the binomial theorem and probability, so deciding whether order matters is the first step every time.

Also taught in Algebra 2, 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.

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

Sigma Prep · sigmaprep.io/worksheets

Name

Permutations and Combinations

Date Period

Evaluate each expression, or count each set of choices.

  1. Decide if order matters, then count

    In how many orders can 3 of 17 runners finish first through third?
  2. A pizza shop offers 17 toppings. How many different pizzas with 5 different toppings can be made?
  3. How many 2-letter codes can be made from 12 different letters if no letter repeats?
  4. In how many ways can 2 of 17 different books be arranged in a row on a shelf?
  5. A test has 16 essay questions and a student answers 4 of them. How many different sets of questions can the student choose?
  6. 10 runners are in a race. In how many ways can the first 2 places be filled?
  7. A club of 15 members chooses a president, a vice president, a secretary, a treasurer and a historian. In how many ways can it be done?
  8. How many different groups of 3 can be chosen from 15 people?
  9. A club of 12 members elects a president, a vice president and a treasurer. How many ways can this be done?
  10. In how many orders can 2 of 16 runners finish first and second?
  11. A test has 17 essay questions and a student answers 4 of them. How many different sets of questions can the student choose?
  12. A club of 9 members chooses a committee of 5. How many committees are possible?
  13. A club of 16 members chooses a president, a vice president and a secretary. In how many ways can it be done?
  14. How many different groups of 2 can be chosen from 15 people?
  15. A teacher picks 5 of 13 students to present. How many groups are possible?

  1. Count arrangements of letters with repeats

    How many different arrangements of the letters in TENNESSEE are there?
  2. How many different arrangements of the letters in CALCULUS are there?
  3. How many different arrangements of the letters in MISSISSIPPI are there?
  4. How many different arrangements of the letters in COOKBOOK are there?
  5. How many different arrangements of the letters in STATISTICS are there?

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

Sigma Prep · sigmaprep.io/worksheets

Name

Permutations and CombinationsAnswer keyVersion 2

Date Period

Evaluate each expression, or count each set of choices.

  1. Decide if order matters, then count

    In how many orders can 3 of 17 runners finish first through third?4080
  2. A pizza shop offers 17 toppings. How many different pizzas with 5 different toppings can be made?6188
  3. How many 2-letter codes can be made from 12 different letters if no letter repeats?132
  4. In how many ways can 2 of 17 different books be arranged in a row on a shelf?272
  5. A test has 16 essay questions and a student answers 4 of them. How many different sets of questions can the student choose?1820
  6. 10 runners are in a race. In how many ways can the first 2 places be filled?90
  7. A club of 15 members chooses a president, a vice president, a secretary, a treasurer and a historian. In how many ways can it be done?360360
  8. How many different groups of 3 can be chosen from 15 people?455
  9. A club of 12 members elects a president, a vice president and a treasurer. How many ways can this be done?1320
  10. In how many orders can 2 of 16 runners finish first and second?240
  11. A test has 17 essay questions and a student answers 4 of them. How many different sets of questions can the student choose?2380
  12. A club of 9 members chooses a committee of 5. How many committees are possible?126
  13. A club of 16 members chooses a president, a vice president and a secretary. In how many ways can it be done?3360
  14. How many different groups of 2 can be chosen from 15 people?105
  15. A teacher picks 5 of 13 students to present. How many groups are possible?1287

  1. Count arrangements of letters with repeats

    How many different arrangements of the letters in TENNESSEE are there?3780
  2. How many different arrangements of the letters in CALCULUS are there?5040
  3. How many different arrangements of the letters in MISSISSIPPI are there?34650
  4. How many different arrangements of the letters in COOKBOOK are there?840
  5. How many different arrangements of the letters in STATISTICS are there?50400

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

How to do these

How many different arrangements of the letters in LETTER are there?

  1. 6 letters give 6! = 720 orders.
  2. E and T each appear twice, so divide by 2! × 2! = 4.
  3. 720 ÷ 4.

The answer is 180.

Where students go wrong

Dividing out only one repeated letter. In MISSISSIPPI the S and the I each appear four times and the P twice, so 11! is divided by 4! × 4! × 2!, giving 34,650.

Also called nPr and nCr.

What you can put on this worksheet

Permutations
P(8,3) → 336
Combinations
(83) → 56
Permutation or combination? Word problems
A teacher picks 2 of 7 students to present. How many groups are possible? → 21
Letters with repeats
How many different arrangements of the letters in SUCCESS are there? → 420
Arrange all of them: n!
4 runners finish a race with no ties. In how many different orders can they finish? → 24
List every arrangement of a small set
List every way to arrange the letters B, M and P. → {BMP, BPM, MBP, MPB, PBM, PMB}
List every combination of a small set
List every group of 2 letters that can be chosen from B, F, M and P. → {BF, BM, BP, FM, FP, MP}
nCr with r close to n, and in expressions
11C9 → 55
Permutation or combination? Say which
A committee of 3 is chosen from 16 members. Is this a permutation or a combination? → Combination
Handshakes, games, locks and other tricky counts
9 teams play in a tournament and the top 4 move on to the next round. How many different groups of teams could move on? → 126
Find n from nPr or nCr
Given that nC2=55, find n. → 11
A combination equal to another, and why
Without working either out, explain why 14C4=14C10 → Choosing the 4 to take is the same as choosing the 10 to leave
nPr with a variable; a word with so many arrangements
Write a word (any letters) whose letters can be arranged in exactly 6 ways → One answer: AABB
Cards to everyone vs handshakes
Every two of 9 people trade phone numbers once. How many trades? → 36
nPr and nCr with a letter
Simplify: nP2+nC2 → 3n(n−1)2

Questions about these worksheets

Yes. Every permutations and combinations 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.

Precalculus is usually taken around 11th or 12th 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 permutations and combinations 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 15: permutations, combinations, permutation or combination? word problems, letters with repeats, arrange all of them: n!, list every arrangement of a small set, list every combination of a small set, ncr with r close to n, and in expressions, permutation or combination? say which, handshakes, games, locks and other tricky counts, find n from npr or ncr, a combination equal to another, and why, npr with a variable; a word with so many arrangements, cards to everyone vs handshakes and npr and ncr with a letter. Tick as many as you want and set how many of each, or let it spread them evenly.

Using permutations when order does not matter. Picking three people for a committee is a combination; picking first, second and third place is a permutation. If swapping two of your choices gives the same outcome, divide.