Combination Calculator
Calculate combinations (nCr), the number of ways to choose r items from n when order does not matter, with exact results and step-by-step working.
How many items are available? e.g. 10 people.
How many are being selected from n? e.g. a 3-person committee.
Combination result
There are 120 ways to choose 3 items from 10 when order does not matter.
Step-by-step calculation
Step 1 — Write the formula
nCr = n! / (r!(n − r)!), with n = 10 and r = 3
C(10, 3) = 10! / (3!(10 − 3)!) = 10! / (3! × 7!)
Step 2 — Calculate the factorial terms
- 10! = 3,628,800
- 3! = 6
- 7! = 5,040
Step 3 — Substitute
C(10, 3) = 3,628,800 / (6 × 5,040) = 3,628,800 / 30,240
Step 4 — Simplify
C(10, 3) = 120
Step 5 — Check with the shortcut
= (10 × 9 × 8) / (3 × 2 × 1) = 720 / 6
The 7! in the denominator cancels the matching factors of 10!, leaving the 3 largest factors on top.
Result
C(10, 3) = 120
Symmetry: C(10, 3) = C(10, 7), because choosing 3 items to include is the same as choosing 7 to leave out.
Combination vs. permutation
| Concept | Combination | Permutation |
|---|---|---|
| Order matters? | No | Yes |
| Formula | n! / (r!(n − r)!) | n! / (n − r)! |
| Example | Choosing 3 committee members | Awarding 1st, 2nd, and 3rd place |
| ABC vs. BAC | Same selection | Different arrangements |
| Your inputs | C(10, 3) = 120 | P(10, 3) = 720 |
If order does not matter, use a combination; if it matters, use a permutation. Each group of 3 can be ordered in 3! = 6 ways, so nPr = nCr × r!.
Need ordered arrangements? Use the Permutation Calculator. To turn a count of combinations into a chance, divide the favorable selections by the total in the Probability Calculator, or work out a single factorial with the Factorial Calculator.
This combination calculator (nCr calculator) counts the number of ways to choose r items from n different items when order does not matter. Enter n and r to get the exact answer, the formula filled in with your numbers, and each step of the calculation, along with the matching permutation count for comparison.
In a combination, choosing A and B is the same selection as choosing B and A, so it is counted once.
Worked Calculation Examples
| Scenario | Result | Calculation Step |
|---|---|---|
| 10C3: choosing a 3-person committee from 10 | 120 | 10! ÷ (3! × 7!) = (10 × 9 × 8) ÷ (3 × 2 × 1) = 720 ÷ 6 = 120. Order does not matter because the committee is the same group whoever is picked first. |
| 6C2: choosing 2 items from 6 | 15 | (6 × 5) ÷ (2 × 1) = 30 ÷ 2 = 15. Picking items A and B is the same pair as picking B and A. |
| 8C4: selecting 4 of 8 applicants | 70 | (8 × 7 × 6 × 5) ÷ (4 × 3 × 2 × 1) = 1,680 ÷ 24 = 70. Only who is selected matters, not the order of selection. |
| 10C7: the same as 10C3 by symmetry | 120 | Choosing 7 to include is choosing 3 to leave out, so 10C7 = 10C3 = 120. |
| 52C5: 5-card hands from a standard deck | 2,598,960 | (52 × 51 × 50 × 49 × 48) ÷ 5! = 311,875,200 ÷ 120 = 2,598,960. A hand is the same whatever order the cards are dealt in. |
The Combination Formula
The number of combinations of r items from n is nCr = n! / (r!(n − r)!), also written C(n, r). Here n is the total number of available items, r is the number selected, n − r is the number left out, and n! (n factorial) is the product of every whole number from n down to 1, with 0! = 1.
Why factorials? n! counts every way to line up all n items. But a combination only cares which items are in the chosen group, not their order, so the r! orderings inside the chosen group and the (n − r)! orderings of the items left out are divided away. For 10C3: 10! / (3! × 7!) = 3,628,800 / (6 × 5,040) = 120.
What Is a Combination?
A combination is a selection where order does not matter. Suppose you pick 2 people from A, B, and C. The possible groups are AB, AC, and BC, so 3C2 = 3. Picking A then B gives the same group as picking B then A, so AB and BA count only once.
If the order did matter, as it would when giving out a first and second prize, there would be 6 outcomes (AB, BA, AC, CA, BC, CB), which is a permutation count.
When Are Combinations Used?
Use combinations whenever you are counting groups rather than arrangements: choosing committee members or a team from a larger group, selecting research participants, counting possible poker hands (52C5 = 2,598,960), counting lottery tickets (choosing 6 numbers from 49 gives 49C6 = 13,983,816), or choosing toppings, menu items, or questions to answer from a list.
Combinations are also a common first step in probability. To find the chance that a random 5-card hand is a particular kind of hand, count the favorable hands with combinations, divide by the total number of hands (52C5), and the result is the probability; the Probability Calculator can do that final division for you.
Combination vs. Permutation
If order does not matter, use a combination; if order matters, use a permutation. Choosing 3 committee members from 10 is a combination (10C3 = 120), while awarding 1st, 2nd, and 3rd place to 3 of 10 runners is a permutation (10P3 = 720). ABC and BAC are the same selection but different arrangements.
The two are linked: every group of r items can be ordered in r! ways, so nPr = nCr × r!. For 10 and 3, 720 = 120 × 6. Use the Permutation Calculator when order matters.
Combination Symmetry
Choosing r items to include is the same as choosing n − r items to leave out, so nCr = nC(n − r). For example, 10C3 = 10C7 = 120: every choice of 3 people for a committee is also a choice of 7 people to leave off it.
This also makes the calculation faster: the calculator always works with the smaller of r and n − r. It explains the edge cases too: nC0 = nCn = 1, because there is exactly one way to choose nothing and exactly one way to choose everything.
How to Use the Combination Calculator
- Enter the total number of items available (n).
- Enter how many items are being selected (r). Both must be whole numbers, and r cannot be greater than n.
- Read the number of combinations. Results update as you type, and very large results show a scientific approximation along with the exact value.
- Follow the step-by-step calculation to see the factorials filled in with your numbers and cancelled down to the answer.
- Compare the result with the permutation count in the table below it, then use Copy result to copy the answer, formula, and inputs, or Reset to start again.
Frequently Asked Questions
What is a combination?
A combination is a selection of items where order does not matter. Choosing A and B is the same combination as choosing B and A.
What is the nCr formula?
nCr = n! / (r!(n − r)!), where n is the total number of items, r is the number selected, and ! means factorial. For 6C2: 6! / (2! × 4!) = 720 / (2 × 24) = 15.
What does nCr mean?
nCr, also written C(n, r) or "n choose r", is the number of ways to choose r items from n different items when order does not matter.
What is the difference between combinations and permutations?
Combinations count selections where order does not matter; permutations count arrangements where it does. From 10 people, there are 10C3 = 120 possible 3-person committees but 10P3 = 720 ways to award gold, silver, and bronze.
Does order matter in combinations?
No. That is what makes it a combination: AB and BA are the same selection and are counted once. If order matters, count permutations instead.
Can r be greater than n?
No. You cannot choose more items than are available, so r must be between 0 and n. (Some textbooks define nCr = 0 when r > n; this calculator asks you to correct the inputs instead.)
What is 10C3?
10C3 = 120. It is 10! / (3! × 7!) = (10 × 9 × 8) / (3 × 2 × 1) = 720 / 6 = 120, the number of different 3-item groups from 10 items.
What is 10C0?
10C0 = 1. There is exactly one way to choose nothing: 10! / (0! × 10!) = 1, since 0! = 1. Likewise 10C10 = 1.
Why is nCr equal to nC(n − r)?
Choosing r items to include is the same as choosing the n − r items to leave out, so both count the same groups. For example, 10C3 = 10C7 = 120.
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Last updated: September 27, 2026.