nCr formula: nCr = n! / (r!(n − r)!)
The nCr formula is nCr = n! / (r!(n − r)!).
What each letter means
- \(n\) how many items there are to choose from
- \(r\) how many you choose
- \(n!\) n factorial: n × (n − 1) × … × 2 × 1, with 0! = 1
When to use it
To count selections where the order does not matter (teams, committees, hands of cards), and for the coefficients in a binomial expansion.
Worked example
In how many ways can a team of 4 be chosen from 10 students?
- \(\binom{10}{4}=\dfrac{10!}{4!\,6!}=\dfrac{10\times9\times8\times7}{4\times3\times2\times1}\)
Answer: \(210\) ways
Common mistake
Using nPr when the order does not matter. Choosing Asha then Ben is the same team as Ben then Asha, so use nCr.
On your course
| Course | In the exam |
|---|---|
| AA SL | In the IB formula booklet |
From our own IB Maths formula sheets, in our words. The IB gives its booklet to schools, so we checked against our own copy: your teacher has the official one. Official: IB DP Mathematics page.
Practise and revise
- Practise AA HL counting and binomial questions
- Revise Binomial expansion (AA SL)
- Print AA SL one-page formula sheet
Questions
What is the nCr formula?
The nCr formula is nCr = n! / (r!(n − r)!). n: how many items there are to choose from; r: how many you choose; n!: n factorial: n × (n − 1) × … × 2 × 1, with 0! = 1.
Is the nCr formula (combinations) given in the exam?
AA SL: in the IB formula booklet. This comes from our own IB Maths formula sheets; your teacher has the official booklet.
What is the difference between nCr and nPr?
nCr counts selections, where order does not matter. nPr = n!/(n − r)! counts arrangements, where it does, so nPr = nCr × r!.
Our own wording, examples and card, checked by IB Math Revision. Not produced or endorsed by the International Baccalaureate Organization.