What Is nCr?
nCr, read "n choose r" and also written C(n, r), counts combinations. The formula is nCr = n! ÷ (r! (n − r)!). Because the order of the chosen items does not matter, nCr is always smaller than or equal to the number of ordered arrangements nPr, by a factor of r!.
Alongside the exact result, the calculator shows the number of orderings of the r items, the symmetry C(n, r) = C(n, n − r) and a check against the permutation count. If you want permutations or repetition as well, the combination and permutation calculators linked below cover them.
Further down you will find a variables table, worked examples and answers to common questions such as how to calculate nCr by hand and why C(n, 0) = 1.
How to Use the nCr Calculator
- Enter n, the number of items to choose from.
- Enter r, the number you are choosing.
- Read C(n, r) and compare with the permutations result if order matters.
- Open the steps to see how the factorials cancel.
nCr Formula
Combination formula
Symmetry
Pascal’s rule
Sum of a row
nCr Variables Explained
| Symbol | Meaning | Notes |
|---|---|---|
| n | Total items | Whole number |
| r | Items chosen | 0 ≤ r ≤ n |
| ! | Factorial | n! = n × (n − 1) × … × 1; 0! = 1 |
| nCr | Number of combinations | Order ignored |
nCr Worked Example
With the example values loaded in the calculator above, the steps are:
- Formula: C(n, r) = n! ÷ (r! × (n − r)!).
- Cancel the common factorial: C(52, 5) = (52 × 51 × 50 × 49 × 48) ÷ (5 × 4 × 3 × 2 × 1).
- Result: 2,598,960.
- Symmetry check: choosing which 5 to take is the same as choosing which 47 to leave, so C(52, 5) = C(52, 47).
Result: C(52, 5) = 52C5 = 2,598,960.
How to Read nCr
C(n, r) counts the ways to choose r items from n when order does not matter. C(52, 5) = 2,598,960, the number of distinct 5-card poker hands. Dividing P(52, 5) by 5! gives the same number, because each hand can be ordered in 120 ways.
Symmetry gives C(n, r) = C(n, n − r), so C(52, 5) = C(52, 47). Each entry of Pascal’s triangle is the sum of the two above it, which the calculator checks with C(51, 4) + C(51, 5).
Common nCr examples
| Situation | Calculation | Number of ways |
|---|---|---|
| 5-card poker hands | C(52, 5) | 2,598,960 |
| Committee of 3 from 10 people | C(10, 3) | 120 |
| US Powerball white balls (5 of 69) | C(69, 5) | 11,238,513 |
| Pick 6 of 49 lottery | C(49, 6) | 13,983,816 |
| Choose 2 teams from 8 to play | C(8, 2) | 28 |
| Subsets of a 10-element set | 2¹⁰ | 1,024 |
nCr vs nPr: combination or permutation?
Ask yourself whether swapping two chosen items gives a different outcome. If it does, such as ranking runners or assigning roles, use permutations nPr. If it does not, such as picking a team or hand of cards, use combinations nCr. The two are linked: nPr = nCr × r!, because each group of r items can be ordered in r! ways.
| Question | Order matters? | Formula |
|---|---|---|
| Pick a team of 4 from 12 | No | nCr |
| Award 1st, 2nd, 3rd place among 12 | Yes | nPr |
| Choose 4 digits for a code, no repeats | Yes | nPr |
| Choose toppings from a list | No | nCr |
nCr in probability
Combinations turn counting into probability. The chance of a specific poker hand is the number of such hands divided by C(52, 5). The binomial distribution multiplies nCr by pᵏ(1 − p)ⁿ⁻ᵏ to count the ways successes can be arranged among the trials. The hypergeometric distribution uses ratios of nCr values for sampling without replacement.
Where nCr Is Used
Card hands and lotteries
Count possible selections.
Teams and committees
Count ways to choose a group from a larger one.
Binomial coefficients
Get the coefficients of (a + b)ⁿ.
Probability
Use as the denominator for equally likely selections.
Common nCr Mistakes to Avoid
- Use combinations when order does not matter.
- C(n, 0) = C(n, n) = 1.
- Cancel factorials before multiplying when you calculate by hand.
- Check that r is not larger than n.
nCr Terms and Definitions
- Binomial coefficient
- Another name for nCr, the coefficient of xʳ in (1 + x)ⁿ.
- Factorial
- n! = n × (n − 1) × … × 1.
nCr FAQs
What is nCr?
The number of ways to choose r items from n without regard to order, equal to n! ÷ (r!(n − r)!).
How do I calculate nCr by hand?
Multiply n down r terms and divide by r factorial. For C(10, 3): (10 × 9 × 8) ÷ (3 × 2 × 1) = 120.
What is the difference between nCr and nPr?
nPr counts ordered arrangements; nCr counts unordered selections. nPr = nCr × r!.
Why is C(n, r) equal to C(n, n − r)?
Choosing r items to take is equivalent to choosing n − r items to leave behind.
What is C(n, 0)?
It is 1: there is exactly one way to choose nothing.
How big can n be?
This calculator accepts n up to 5,000 and shows exact integers or scientific notation for very large results.