Combinations is one of 2 counting formulas in the probability section of this library, and it is used at high school · ap level.
Why combinations works
Every unordered group of r items was counted many times among the ordered selections, once for each way to arrange those r items. Since there are r factorial such arrangements, dividing the ordered count by r factorial removes the duplication exactly.
What each symbol means
$n$ is population size and $r$ selection size.
Combinations: when it holds
$0\le r\le n$ are integers and order does not matter.
When it stops applying
It assumes distinguishable items chosen without replacement, and for probability it also assumes every group is equally likely. Allowing repeats changes the count entirely: three scoops from five flavours with repetition gives 35 possibilities rather than 10.
Combinations: a worked example
Choosing 3 of 5 people gives $\binom53=10$.
The mistake to avoid
What people do: Leaving out the division by r factorial when order does not matter.
Why it goes wrong: Picking 3 people from 5 as an unordered group gives 10, but skipping the division gives 60. Every committee would be counted six times, once for each way to line its members up.
Do this instead: After computing the ordered count, divide by the factorial of how many items you chose, not by the factorial of how many were available.
Combinations: step by step
- Name the unknown, and the unit the answer has to come out in.
- Match the symbols to your values. $n$ is population size and $r$ selection size.
- Check the conditions before substituting. $0\le r\le n$ are integers and order does not matter.
- Substitute, keep exact values to the last line, then test the sign, size, and unit against a rough estimate — the check that catches most probability slips.
Where this formula fits
- Subject
- Probability formulas — 17 entries in this library
- Topic
- Counting
- Level
- High school · AP
Formulas are easiest to keep when they sit inside a method rather than on a list. Use the links below to see where combinations comes from, to check a calculation against a tool, and to practise it until you can recall it without looking.
- Probability Foundations — the lesson behind this formula: model outcomes, complements, and conditional information.
- Statistics Calculator — check your substitution and the value it produces.
- Study probability — the subject guide that explains the ideas these formulas compress.
- Statistics Practice — questions that make you retrieve the formula instead of recognising it.
- All 17 probability formulas — the full grouped reference, or the complete formula library.
Questions about combinations
Why does choosing r equal choosing n minus r?
Because picking who is in is the same as picking who is out. Choosing 3 from 10 and choosing 7 from 10 both give 120, since each selection determines the other.
How does this connect to Pascal's triangle?
Each entry in the triangle is one of these counts, and the rule that builds each one from the pair diagonally above it reflects a selection either containing a particular item or leaving it out.
What is the value when r is zero?
It is 1. There is exactly one way to choose nothing, namely the empty selection, which is why every row of Pascal's triangle starts with a 1.
How many five-card poker hands are there?
There are 2598960, which is choosing 5 from 52. That single number is the denominator behind almost every poker probability you will ever compute.