About the Combination Calculator
What is the Combination Calculator?
A combination is a selection in which order does not matter. Choosing 3 people for a committee from 10, or 6 lottery numbers from 49, are combinations: the group {A, B, C} is the same however it is listed.
Enter n and r. Without repetition each item can be chosen at most once; with repetition an item can be picked several times, like choosing 3 scoops from 5 ice-cream flavours where two scoops can be the same flavour.
How to use the Combination Calculator
- Total Items (n) (sample: 10)
- Items Chosen (r) (sample: 3)
- Repetition — allow repetition when the same item can be chosen more than once. (sample: Without repetition)
- Press Calculate Combinations (results also refresh as you type). The result card shows the Number of Combinations (nCr), along with Formula used and Order matters.
Formula used
nCr = n! ÷ (r! × (n − r)!)
- n = the number of items to choose from
- r = the number of items chosen
- With repetition = (n + r − 1)C r
nCr = nPr ÷ r!, because each group of r items can be ordered in r! ways. The calculator multiplies step by step so it never overflows on the factorials.
Worked example
These are the sample values already filled in the calculator above, and the result it returns for them:
| Input | Sample value |
|---|---|
| Total Items (n) | 10 |
| Items Chosen (r) | 3 |
| Repetition | Without repetition |
| Result | Value |
|---|---|
| Number of Combinations (nCr) | 120 |
| Formula used | n! / (r! (n − r)!) |
| Order matters | No |
Tips for using the Combination Calculator
- nCr = nC(n − r): choosing 3 people to include is the same as choosing 7 to leave out.
- The odds of winning a 6-from-49 lottery are 1 in 49C6 = 13,983,816.
- The numbers nCr form Pascal's triangle and are the coefficients in the binomial expansion of (a + b)ⁿ.