Binomial Coefficient Calculator
Enter n and k to find the binomial coefficient, “n choose k”.
Updated
Binomial Coefficient Calculator
Our Binomial Coefficient Calculator finds “n choose k”, the number of ways to pick k items from a set of n items when order doesn’t matter. It’s the same value the nCr button on a scientific calculator produces, but exact for very large numbers, with the working shown as you type.
Quick answer
The binomial coefficient counts combinations: C(n, k) = n! ÷ (k! × (n − k)!). For example, C(6, 2) = 720 ÷ (2 × 24) = 15, so there are 15 ways to choose 2 items from 6. Enter any n and k above and the calculator shows the result instantly.
- Also written nCr, nCk or “n choose k”
- Order doesn’t matter: AB and BA count as one combination
- Well-known values: C(52, 5) = 2,598,960 poker hands; C(49, 6) = 13,983,816 lottery tickets
- Edge cases: C(n, 0) = C(n, n) = 1, and C(n, k) = 0 when k > n
What Is a Binomial Coefficient?
The binomial coefficient, written C(n, k) or “n choose k”, counts how many distinct groups of k items can be formed from a set of n items. It is called “binomial” because the same numbers appear as the coefficients when you expand (x + y)n: for example, (x + y)³ = x³ + 3x²y + 3xy² + y³, and 1, 3, 3, 1 are C(3, 0) to C(3, 3).
Binomial Coefficient Formula
C(n, k) = (n × (n − 1) × … × (n − k + 1)) ÷ k!
Here n! (n factorial) is the product of all whole numbers from 1 to n; see the Factorial Calculator. Both n and k are whole numbers of 0 or more. The second line is quicker by hand: multiply k numbers counting down from n, then divide by k!. Choosing k items is the same as leaving out n − k, so C(n, k) = C(n, n − k).
Example Calculations
C(10, 3) = (10 × 9 × 8) ÷ 3! = 720 ÷ 6 = 120
C(52, 5) = (52 × 51 × 50 × 49 × 48) ÷ 5! = 2,598,960
C(49, 6) = 13,983,816
There are 15 ways to choose 2 items from 6, 120 ways to pick a 3-person committee from 10 people, 2,598,960 possible 5-card poker hands and 13,983,816 possible 6/49 lottery tickets.
Pascal’s Triangle
Every binomial coefficient appears in Pascal’s triangle. Row n lists C(n, 0) to C(n, n), and each number is the sum of the two above it:
n = 1: 1 1
n = 2: 1 2 1
n = 3: 1 3 3 1
n = 4: 1 4 6 4 1
n = 5: 1 5 10 10 5 1
So C(5, 2) = 10 is the third number in row 5, and it equals C(4, 1) + C(4, 2) = 4 + 6. The numbers in row n add up to 2n.
Key Terms
- n
- The total number of different items to choose from.
- k
- How many items are chosen.
- Combination
- A selection where order doesn’t matter. C(n, k) counts them.
- Permutation
- An arrangement where order matters: nPr = C(n, k) × k!.
- Factorial (n!)
- The product of all whole numbers from 1 to n, with 0! = 1.
How to Use the Binomial Coefficient Calculator
- Enter n. Type the total number of items in the “Total items (n)” box, for example 52 for a deck of cards. Pressing Enter moves you to the next box.
- Enter k. Type how many items are chosen in the “Items chosen (k)” box, for example 5 for a poker hand.
- Read n choose k. The result appears as soon as both boxes are filled, with the working underneath (for example C(6, 2) = 6! ÷ (2! × 4!) = 720 ÷ (2 × 24) = 15). Results longer than 21 digits are shown in short form with the number of digits.
- Copy the result. Press Copy to copy the exact result with every digit. Press Clear or Esc to start again.
Example: enter n = 52 and k = 5 and the result is 2,598,960, with the working (52 × 51 × 50 × 49 × 48) ÷ 5!.
Who Can Use This Calculator?
Frequently Asked Questions (FAQs)
What is a binomial coefficient?
The binomial coefficient C(n, k), read “n choose k”, is the number of ways to choose k items from n different items when the order doesn’t matter. For example, C(6, 2) = 15: there are 15 different pairs you can pick from 6 items.
What is the formula for n choose k?
C(n, k) = n! ÷ (k! × (n − k)!), where ! means factorial. For C(6, 2): 6! ÷ (2! × 4!) = 720 ÷ (2 × 24) = 15. A quicker way for small k is to multiply k numbers counting down from n and divide by k!: (6 × 5) ÷ 2 = 15.
Is the binomial coefficient the same as nCr?
Yes. C(n, k), nCr, nCk and “n choose k” are all the same number: the count of combinations. The nCr key on a scientific calculator gives the same result as this calculator.
What is the difference between combinations and permutations?
Combinations ignore order; permutations count every order separately. Choosing 2 of 3 letters A, B, C gives 3 combinations (AB, AC, BC) but 6 permutations (AB, BA, AC, CA, BC, CB). In general, permutations = combinations × k!, so nPr = C(n, k) × k!.
How many 5-card poker hands are there?
C(52, 5) = (52 × 51 × 50 × 49 × 48) ÷ 5! = 2,598,960. That is the number of different 5-card hands from a standard 52-card deck, since the order the cards are dealt in doesn’t matter.
What are the odds of winning a 6/49 lottery?
There are C(49, 6) = 13,983,816 ways to choose 6 numbers from 49, so a single ticket has a 1 in 13,983,816 chance of matching all six.
What is C(n, 0), C(n, n) and C(n, k) when k is larger than n?
C(n, 0) = 1 and C(n, n) = 1, because there is exactly one way to choose nothing and one way to choose everything. When k is larger than n, C(n, k) = 0, because you can’t choose more items than there are. The calculator shows 0 in that case.
How is the binomial coefficient related to Pascal’s triangle?
Row n of Pascal’s triangle lists C(n, 0), C(n, 1), …, C(n, n). Row 5 is 1, 5, 10, 10, 5, 1. Each entry is the sum of the two above it, C(n, k) = C(n − 1, k − 1) + C(n − 1, k), and each row adds up to 2n.
Is the Binomial Coefficient Calculator free?
Yes. It is completely free, needs no sign-up, and can be used as many times as you like.
Try our Binomial Coefficient Calculator to find “n choose k” in seconds.