This permutation combination calculator focuses on combinations: how many ways to choose r items from n when order does not matter. Enter n and r to get nCr = n! ÷ (r! × (n-r)!). The default inputs n = 10 and r = 3 return 120.
Teachers and contest prep students use it to check counting problems quickly. For list averages after you collect scores, see the mean median mode range calculator.
How the formula works
Factorials grow fast, so the tool evaluates the choose formula carefully for your integers. Combinations ignore order: choosing A then B is the same pair as B then A. That is why nCr is smaller than the matching permutation count when r is greater than 1.
Worked example
n = 10, r = 3. C(10,3) = 10×9×8 ÷ (3×2×1) = 120.
| Input | Value |
|---|---|
| n | 10 |
| r | 3 |
| nCr | 120 |
How to use the fields
- n is the size of the full set.
- r is how many items you select.
Combinations versus permutations
If a problem says “committee of three,” order usually does not matter, so use combinations. If it says “first, second, and third place,” order matters, so permutations are the right count. Read the wording before you trust a number.
Common mistakes
- Using nCr when the problem ranks ordered outcomes
- Setting r larger than n
- Forgetting that C(n,r) equals C(n, n-r)
- Mixing probability percentages with raw counts
Classroom checklist
- Underline whether order matters in the problem statement.
- Confirm n and r are integers.
- Compute nCr, then interpret the count in a sentence.
- If the question asks for probability, divide by the total outcomes next.
Symmetric identity
Choosing 3 from 10 is the same count as leaving out 7 from 10, so C(10,3) = C(10,7). That identity helps you pick the smaller r for hand calculation and spot typos when tools disagree.
Large n caution
Huge n values make factorials enormous. Stay within sensible homework and planning ranges. If a result looks impossible for the story problem, recheck that you did not swap n and r.
Story problems that need nCr
Choosing 3 starters from 10 players for a photo, when order in the photo line does not matter, is C(10,3). Choosing 3 toppings from 10 when the pizza does not care about topping order is the same structure. If the problem instead assigns captain, vice captain, and secretary as distinct roles, you need permutations instead.
Write a one line reason: “order matters” or “order does not matter.” That sentence prevents most counting errors before you touch factorials.
Building intuition for 120
With n=10 and r=3, you can think of 10 choices for the first pick, 9 for the second, and 8 for the third, which is 720 ordered triples. Because order does not matter for combinations, divide by 3! = 6 arrangements of each triple. 720 ÷ 6 = 120, matching C(10,3).
That bridge between permutations and combinations helps you sanity check large answers. If your combination result is larger than the related permutation count, something is reversed.
Contest and homework habits
Circle n and r in the problem text. Confirm r is not larger than n. Compute nCr, then reread the question to see whether it asked for a count or a probability. Probabilities need a denominator for total outcomes after you have the favorable count.
When two methods disagree, recompute C(n,r) and C(n,n-r). They must match. Disagreement means an arithmetic slip, not a philosophical debate about counting.
Limits of hand calculation
Factorials grow quickly. Hand work is fine for small classroom n values like the default. For large n, rely on the calculator and keep the story interpretation clear so you do not paste a huge number into the wrong sentence.
Probability follow through
If every committee of 3 from 10 is equally likely, the probability of one specific committee is 1 ÷ 120. That step only works after you trust the combination count. Mixing nCr into a percent without a clear total outcomes denominator is a common test trap.
When outcomes are not equally likely, counting with nCr is still useful for listing options, but probability needs weights. Keep those ideas separate in your write up.
Limitations
The primary result reported for this page’s engine check is nCr. Story problems about ordered lists need a permutation step as well. Probability still needs a separate model after you have the count.