What is the difference between a permutation and a combination?

The difference is whether swapping two chosen items gives a new result. In a permutation it does, as in a door code, where 1-2-3 and 3-2-1 are two different codes. In a combination it does not, as in a hand of cards. From 4 letters, ₄P₃ = 24 but ₄C₃ = 4, because each group of 3 letters can be lined up in 6 orders.

How do you work out combinations without a calculator?

Count the ordered picks, then divide by the number of ways to order each group. For ₇C₃, there are 7 × 6 × 5 = 210 ordered picks of three. Each group of three can be ordered in 3 × 2 × 1 = 6 ways, so ₇C₃ = 210 ÷ 6 = 35. Cancelling first is faster: the 6 on top cancels the 6 below, leaving 7 × 5 = 35.

Is there a shortcut for finding nCr?

Yes, Pascal’s triangle holds every combination. Start with 1 at the top and write each new number as the sum of the two above it. Row 6, counting the top row as row 0, reads 1, 6, 15, 20, 15, 6, 1, so ₆C₃ = 20. The row is symmetric, so ₆C₄ = ₆C₂ = 15: choosing 4 to keep is the same as choosing 2 to leave out.

Why do students mix up permutations and combinations?

Students mix them up because the two questions often sound alike, and the words do not always say whether order counts. Listing a small case settles it. From A, B and C, the pairs are AB, AC and BC, so ₃C₂ = 3. The ordered picks are those three plus BA, CA and CB, so ₃P₂ = 6. Seeing both lists side by side makes the difference stick.

Is there a free way to bring combinatorics into a lesson?

There is: all three levels, the Olympic mix included, open free in any browser, with no student accounts or installs. For counting practice at the board, take 5 volunteers and ask how many committees of 3 they could form. Then ask how many ways they could fill chair, secretary and treasurer. The answers, 10 and 60, show each committee lining up in 6 orders. On the schools plan, priced per student, a teacher dashboard follows each student’s progress.

Math practice · reimagined

Combinatorics Game

This combinatorics challenge never has more than 8 items to choose from, yet its answers run from 4 to 1680, the value of ₈P₄ at Olympic level. Settle each count before your rival, and every correct one lands as a blow from your fighter.

Ages 4–99+ · no download · plays in your browser · 23 languages

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What are permutations and combinations?

Permutations and combinations are two ways to count choices without listing them all. A combination is a group where order does not matter: apple and pear make the same pair as pear and apple. A permutation is an ordered line-up, so eating the apple first and then the pear is different from the other way round.

Every level of Combinatorics

Example: how many pairs can you pick from 6?

The symbol ₆C₂ asks how many different pairs you can choose from 6 things when order does not matter.
Picking in order first gives 6 choices for the first item and 5 for the second, so 6 × 5 = 30 ordered picks.
Each pair was counted twice, once as A then B and once as B then A, so halve 30 and type 15.

Learn it step by step

nCk counts how many ways to CHOOSE k things from n when the order doesn't matter — like picking a team, where who's named first makes no difference.

5C2 ("choose 2 from 5") = 10. The formula is n! ÷ (k! × (n−k)!), but the idea is simply: how many different groups can you form?

Type the number of possible groups.

Tips and common mistakes

In ₆P₃, the small 3 tells you how many numbers to multiply, counting down from 6: 6 × 5 × 4 = 120. Stop there, not at 1.
A common mistake is counting both orders in a combination. For ₄C₂, listing AB and BA separately gives 12 picks; cross out the repeats and 6 pairs remain.
At a family meal, ask how many handshakes happen if everyone greets everyone else once. Seven people make 21, which is ₇C₂, and every new guest adds more.

How to play

Each round shows a symbol such as ₇C₃ or ₄P₂: the small left number counts the items, and the right one counts your picks. Type the result on the keypad, and check the letter first, since the Olympic level mixes C and P. Only the first correct count lands, and Broccoli hits harder the sooner it arrives.

— Quick answers —

Combinatorics: the short FAQ

What is the difference between a permutation and a combination?: The difference is whether swapping two chosen items gives a new result. In a permutation it does, as in a door code, where 1-2-3 and 3-2-1 are two different codes. In a combination it does not, as in a hand of cards. From 4 letters, ₄P₃ = 24 but ₄C₃ = 4, because each group of 3 letters can be lined up in 6 orders.
How do you work out combinations without a calculator?: Count the ordered picks, then divide by the number of ways to order each group. For ₇C₃, there are 7 × 6 × 5 = 210 ordered picks of three. Each group of three can be ordered in 3 × 2 × 1 = 6 ways, so ₇C₃ = 210 ÷ 6 = 35. Cancelling first is faster: the 6 on top cancels the 6 below, leaving 7 × 5 = 35.
Is there a shortcut for finding nCr?: Yes, Pascal’s triangle holds every combination. Start with 1 at the top and write each new number as the sum of the two above it. Row 6, counting the top row as row 0, reads 1, 6, 15, 20, 15, 6, 1, so ₆C₃ = 20. The row is symmetric, so ₆C₄ = ₆C₂ = 15: choosing 4 to keep is the same as choosing 2 to leave out.
Why do students mix up permutations and combinations?: Students mix them up because the two questions often sound alike, and the words do not always say whether order counts. Listing a small case settles it. From A, B and C, the pairs are AB, AC and BC, so ₃C₂ = 3. The ordered picks are those three plus BA, CA and CB, so ₃P₂ = 6. Seeing both lists side by side makes the difference stick.
Is there a free way to bring combinatorics into a lesson?: There is: all three levels, the Olympic mix included, open free in any browser, with no student accounts or installs. For counting practice at the board, take 5 volunteers and ask how many committees of 3 they could form. Then ask how many ways they could fill chair, secretary and treasurer. The answers, 10 and 60, show each committee lining up in 6 orders. On the schools plan, priced per student, a teacher dashboard follows each student’s progress.

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