Combinations
Combinations count selections where order does not matter.
About this topic
A combination is a selection where order does not matter. The number of ways to choose r items from n is nCr = n!/(r!(n − r)!), also written C(n, r) or "n choose r". It equals nPr divided by r! because each group is counted r! times as a permutation.
Choosing a committee of 3 from 10 people gives 10C3 = 120 groups, whereas choosing a president, treasurer and secretary would give 720 because roles make order matter. Combinations are symmetric, so nCr = nC(n − r); choosing 3 to include is the same as choosing 7 to leave out. They also appear as the entries of Pascal's triangle and the coefficients in binomial expansions.
Example: the chance of winning a lottery that picks 6 numbers from 49 is 1 in 49C6 = 13,983,816. In poker, the number of 5-card hands is 52C5 = 2,598,960, and the chance of being dealt a royal flush is 4 in that number.
Combinations are used in statistics, genetics, team selection and probability. The main mistake is using permutations when order does not matter, which overcounts. Also remember that choosing none or all gives one way: nC0 = nCn = 1.
Key ideas
- nCr = n!/(r!(n − r)!)
- Pascal's triangle lists them
- Choose 2 from 4: 6
Quick check
Use cases: where Combinations helps
- Homework: review the key idea behind Combinations before you start a problem set.
- Exam revision: use the quick check to test whether you can recall Combinations without notes.
- Visual learning: play with the interactive lab to see how Combinations changes when you adjust its inputs.
- Teaching a friend: explain Combinations in your own words, then compare your version with the key ideas.
Number and algebra skills are used in science, engineering, budgeting and computing.
Frequently asked questions
What is Combinations?
Combinations count selections where order does not matter.
What will I learn in Combinations?
nCr = n!/(r!(n − r)!). Pascal's triangle lists them. Choose 2 from 4: 6.
Is Combinations part of Mathematics?
Yes. It is one of 200 Mathematics topics on LearnViz HighSchoolEdu, with 3 key ideas, a live animation and a quick check.
Does the Combinations lab work on a phone or tablet?
Yes. The animation resizes to fit your screen and supports touch, so you can drag, tap and slide with your finger.
How should I study Combinations?
Read the summary and key ideas first, play with the interactive lab to see the idea in motion, then answer the quick check. If you get a question wrong, go back to the key idea it tests.
Do I need an account, and is my progress saved?
No account is needed. Marking a topic as learned is saved in your own browser, so it stays on that device and is not sent to a server.