Your result will appear here
Fill in the fields and press Calculate.
A combination counts selections where "order doesn't matter": choosing r items from n, it makes no difference which order you pick them — {A, B} and {B, A} are the same selection. From a lottery ticket to forming a committee, from picking pizza toppings to card-hand probability, the answer to "how many different groups" is a combination.
Enter the number of items (n) and how many to choose (r), and C(n, r) is calculated instantly.
Combination Calculator: step-by-step worked example
A 6/49 lottery: choose 6 numbers out of 49, order doesn’t matter. How many tickets are possible?
- Total items (n)
- 49
- Chosen (r)
- 6
- 1
Set up the formula
C(n, r) = n! ÷ (r! × (n − r)!)
= 49! ÷ (6! × 43!)
- 2
Cancel down: ordered selections
n × (n − 1) × … × (n − r + 1)
49 × 48 × 47 × 46 × 45 × 44 = 10,068,347,520
- 3
Remove orderings: divide by r!
C(n, r) = [n × … × (n − r + 1)] ÷ r!
10,068,347,520 ÷ 720 = 13,983,816
- 4
Chance with a single ticket
P = 1 ÷ C(n, r)
= 1 in 13,983,816
- C(49, r):1 → 13,983,816
Good to know: Order doesn’t matter: {5, 12, 33} and {33, 5, 12} are the same ticket. Symmetry shortens the work: C(49, 43) = C(49, 6).
Combination Calculator — how it works
Formula
C(n, r) = n! ÷ [r! × (n−r)!]. Because order doesn't matter, we divide the permutation by r! (counting each group's internal orderings only once).
Combination or permutation?
The single question: does order matter? - If it does NOT → combination (team selection, lottery, a bouquet). - If it does → permutation (race positions, a password, president-and-deputy).
That is why C(n,r) ≤ P(n,r) always; the combination is smaller because it collapses orderings into one.
Useful properties
- C(n, 0) = C(n, n) = 1 (there is one way to choose nothing or everything).
- C(n, r) = C(n, n−r) (choosing r is the same count as choosing the n−r you leave behind).
Where it is used
- Games of chance: how many ways to pick 6 numbers from 49 (lottery odds).
- Grouping: selecting a committee, team or jury from a class.
- Probability: card-hand combinations, quality-control samples.
For ordered selections use a permutation calculator; for the factorial underneath use a factorial calculator.
Worked example
A committee of 3 is to be chosen from 10 students: since order doesn't matter (who was picked first is irrelevant), this is a combination. C(10, 3) = 10! ÷ (3! × 7!) = (10 × 9 × 8) ÷ (3 × 2 × 1) = 720 ÷ 6 = 120 different committees. If instead they were chosen as chair-deputy-secretary (ordered), the answer would be P(10,3) = 720 — which shows why the combination is exactly one-sixth (3! = 6): each committee has 6 internal orderings, and the combination counts them once. In the lottery case, C(49, 6) = 13,983,816 different tickets.
Combination table C(n, r)
| n | r = 1 | r = 2 | r = 3 | r = 4 | r = 5 |
|---|---|---|---|---|---|
| 5 | 5 | 10 | 10 | 5 | 1 |
| 6 | 6 | 15 | 20 | 15 | 6 |
| 7 | 7 | 21 | 35 | 35 | 21 |
| 8 | 8 | 28 | 56 | 70 | 56 |
| 9 | 9 | 36 | 84 | 126 | 126 |
| 10 | 10 | 45 | 120 | 210 | 252 |
| 12 | 12 | 66 | 220 | 495 | 792 |
| 15 | 15 | 105 | 455 | 1,365 | 3,003 |
| 20 | 20 | 190 | 1,140 | 4,845 | 15,504 |
| 30 | 30 | 435 | 4,060 | 27,405 | 142,506 |
| 49 | 49 | 1,176 | 18,424 | 211,876 | 1,906,884 |
Frequently asked questions
How is a combination calculated?+
With C(n, r) = n! ÷ [r! × (n−r)!] — the number of ways to choose r items from n without regard to order. Just enter n and r.
What is the difference between a combination and a permutation?+
In a combination order doesn't matter ({A,B}={B,A}); in a permutation it does. So C(n,r) = P(n,r) ÷ r!, and a combination is always less than or equal to a permutation.
Are lottery odds found with combinations?+
Yes — choosing 6 numbers from 49 is order-independent, so it is a combination: C(49,6) = 13,983,816. The odds of the jackpot are the reciprocal (1 in 13,983,816).
Why is C(n, 0) equal to 1?+
There is exactly one way to choose no items (the empty set). Similarly C(n, n) = 1: there is one way to choose all of them.
What does C(n, r) = C(n, n−r) mean?+
Choosing r items is the same count as choosing the n−r you leave out. For example C(10,3) = C(10,7) = 120 — a symmetry that simplifies calculation.
Which tool do I use if order matters?+
If order matters (race finish, password, titled roles), you need a permutation. Use a permutation calculator to find P(n, r).
Related guides
About this page
- Publisher:
- Kalkoro — Teknolojig Bilişim Teknolojileri A.Ş., İstanbul
- Written by:
- Ahmet Tur — Editor (Math, date & time, unit conversion, physics & chemistry, everyday tools)
- Reviewed by:
- Kübra Yıldız — Editor
Parameters come from official sources; the last update date is shown at the bottom of the page.
Spotted a wrong result? Use the "Report a bug" button at the top of the page; every report is reviewed.
How we calculate and our accuracy policy →
Results are for information only. For official matters the relevant authority's calculation prevails.
Last updated: