Without working either out, explain why _14C_4 = _14C_10

Choosing the 4 to take is the same as choosing the 10 to leave

Permutations and combinations worksheet for Statistics

Statistics uses counting to size a sample space: committees and random groups are combinations, while ordered results such as rankings or seat assignments are permutations. Repeated letters in an arrangement divide out the orders that look the same.

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

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Sigma Prep · sigmaprep.io/worksheets Name Permutations and Combinations Date Period Evaluate each expression, or count each set of choices. Count permutations $_{16} P_{2}$ $_{16} P_{3}$ $P (11, 4)$ $P (14, 3)$ $_{15} P_{2}$ Count combinations $C (9, 4)$ $(\binom{15}{2})$ $(\binom{16}{3})$ $_{15} C_{4}$ $(\binom{12}{2})$ Decide if order matters, then count A pizza place offers 11 toppings. How many pizzas with 4 different toppings are possible? How many 2-letter codes can be made from 12 different letters if no letter repeats? A club of 9 members chooses a committee of 2. How many committees are possible? A teacher picks 2 of 7 students to present. How many groups are possible? 11 runners are in a race. In how many ways can the first 2 places be filled? Count arrangements of letters with repeats How many different arrangements of the letters in CALCULUS are there? How many different arrangements of the letters in TATTOO are there? How many different arrangements of the letters in MISSISSIPPI are there? How many different arrangements of the letters in LETTER are there? How many different arrangements of the letters in ALGEBRA are there? 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.

Count permutations
$_{16} P_{2}$
$_{16} P_{3}$
$P (11, 4)$
$P (14, 3)$
$_{15} P_{2}$

Count combinations
$C (9, 4)$
$(\binom{15}{2})$
$(\binom{16}{3})$
$_{15} C_{4}$
$(\binom{12}{2})$

Decide if order matters, then count
A pizza place offers 11 toppings. How many pizzas with 4 different toppings are possible?
How many 2-letter codes can be made from 12 different letters if no letter repeats?
A club of 9 members chooses a committee of 2. How many committees are possible?
A teacher picks 2 of 7 students to present. How many groups are possible?
11 runners are in a race. In how many ways can the first 2 places be filled?

Count arrangements of letters with repeats
How many different arrangements of the letters in CALCULUS are there?
How many different arrangements of the letters in TATTOO are there?
How many different arrangements of the letters in MISSISSIPPI are there?
How many different arrangements of the letters in LETTER are there?
How many different arrangements of the letters in ALGEBRA are there?



Sigma Prep · sigmaprep.io/worksheets Name Permutations and CombinationsAnswer keyVersion 4 Date Period Evaluate each expression, or count each set of choices. Count permutations $_{16} P_{2}$ $240$ $_{16} P_{3}$ $3360$ $P (11, 4)$ $7920$ $P (14, 3)$ $2184$ $_{15} P_{2}$ $210$ Count combinations $C (9, 4)$ $126$ $(\binom{15}{2})$ $105$ $(\binom{16}{3})$ $560$ $_{15} C_{4}$ $1365$ $(\binom{12}{2})$ $66$ Decide if order matters, then count A pizza place offers 11 toppings. How many pizzas with 4 different toppings are possible? $330$ How many 2-letter codes can be made from 12 different letters if no letter repeats? $132$ A club of 9 members chooses a committee of 2. How many committees are possible? $36$ A teacher picks 2 of 7 students to present. How many groups are possible? $21$ 11 runners are in a race. In how many ways can the first 2 places be filled? $110$ Count arrangements of letters with repeats How many different arrangements of the letters in CALCULUS are there? $5040$ How many different arrangements of the letters in TATTOO are there? $60$ How many different arrangements of the letters in MISSISSIPPI are there? $34650$ How many different arrangements of the letters in LETTER are there? $180$ How many different arrangements of the letters in ALGEBRA are there? $2520$ Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

Permutations and CombinationsAnswer keyVersion 4

Date Period

Evaluate each expression, or count each set of choices.

Count permutations
$_{16} P_{2}$ $240$
$_{16} P_{3}$ $3360$
$P (11, 4)$ $7920$
$P (14, 3)$ $2184$
$_{15} P_{2}$ $210$

Count combinations
$C (9, 4)$ $126$
$(\binom{15}{2})$ $105$
$(\binom{16}{3})$ $560$
$_{15} C_{4}$ $1365$
$(\binom{12}{2})$ $66$

Decide if order matters, then count
A pizza place offers 11 toppings. How many pizzas with 4 different toppings are possible? $330$
How many 2-letter codes can be made from 12 different letters if no letter repeats? $132$
A club of 9 members chooses a committee of 2. How many committees are possible? $36$
A teacher picks 2 of 7 students to present. How many groups are possible? $21$
11 runners are in a race. In how many ways can the first 2 places be filled? $110$

Count arrangements of letters with repeats
How many different arrangements of the letters in CALCULUS are there? $5040$
How many different arrangements of the letters in TATTOO are there? $60$
How many different arrangements of the letters in MISSISSIPPI are there? $34650$
How many different arrangements of the letters in LETTER are there? $180$
How many different arrangements of the letters in ALGEBRA are there? $2520$

How to do these

A committee of 3 is chosen from 10 people. How many committees are possible?

Order does not matter in a committee, so use a combination.
10C3 = (10 × 9 × 8) ÷ (3 × 2 × 1).

The answer is $120$ .

Where students go wrong

Using a permutation for a group. Choosing 3 people for a committee ignores order, so it is 10C3 = 120, not 10P3 = 720.

Also called nPr and nCr.

What you can put on this worksheet

Permutations: $P (8, 3)$ → $336$
Combinations: $(\binom{8}{3})$ → $56$
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: $_{11} C_{9}$ → $55$
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 $_{n} C_{2} = 55$ , find $n$ . → $11$
A combination equal to another, and why: Without working either out, explain why $_{14} C_{4} =_{14} C_{10}$ → 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: $_{n}$ P $_{2} +_{n}$ C $_{2}$ → $\frac{3 n (n - 1)}{2}$

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.

Statistics 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.

Permutations and combinations worksheet for Statistics

Select your difficulty

Count permutations

Count combinations

Decide if order matters, then count

Count arrangements of letters with repeats

Count permutations

Count combinations

Decide if order matters, then count

Count arrangements of letters with repeats

How to do these

Where students go wrong

What you can put on this worksheet

Questions about these worksheets

Are these permutations and combinations worksheets free?+

Do the permutations and combinations worksheets come with an answer key?+

Are these permutations and combinations worksheets printable?+

What grade are these permutations and combinations worksheets for?+

Can I get a different version of the same permutations and combinations worksheet?+

Is this a free alternative to Kuta Software for permutations and combinations?+

Can I choose how hard the permutations and combinations questions are?+

What kinds of question are on a permutations and combinations worksheet?+

What do students get wrong with permutations and combinations?+