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Binomial Coefficient Calculator

Enter n and k to compute the number of combinations and permutations.

How it is calculated

The tool computes C(n, k) = n! ÷ (k! × (n − k)!), the number of ways to choose k items from n when order does not matter, and P(n, k) = C(n, k) × k!, the number of ordered arrangements. To avoid giant factorials it multiplies and divides step by step using the smaller of k and n − k. Inputs are rounded to the nearest whole number (5.6 is treated as 6) and must not be negative; k above 1,000,000 returns no result. If k is greater than n, both results are 0. Values from 1015 upward are shown in scientific notation with five significant figures (for example 1.0089 × 10²⁹), which are floating-point approximations, not exact integers.

Worked example

How many different 5-card hands can be dealt from a 52-card deck? Here n = 52 and k = 5. C(52, 5) = (52 × 51 × 50 × 49 × 48) ÷ (5 × 4 × 3 × 2 × 1) = 311,875,200 ÷ 120 = 2,598,960 hands. The permutation count is 2,598,960 × 120 = 311,875,200, the number of ordered ways to deal five cards in sequence, where the same five cards in a different order count as different outcomes.

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Common mistakes

  • Using the permutation figure when order does not matter. For a lottery ticket or a poker hand, C(n, k) is the right count; P(n, k) is 120 times larger here because it counts each hand in all orders.
  • Swapping n and k. n is the total pool and k is how many you pick, and the tool returns 0 whenever k exceeds n, which is a signal that the two are the wrong way round.
  • Typing a decimal such as 5.6 and expecting an error. The tool rounds it to 6 and gives a result, so enter whole numbers.
  • Treating very large results as exact. Above about 1015 the tool shows a rounded scientific-notation value, so use exact integer software if you need every digit.

About this calculator

The binomial coefficient C(n, k) = n! / (k!(n−k)!), read “n choose k,” counts the number of ways to pick an unordered subset of k items from a set of n — for example, C(5, 2) = 10 ways to choose 2 people from a group of 5. This calculator also reports the related permutation count P(n, k) = n! / (n−k)!, which counts the same selection but where order matters, so P(5, 2) = 20 since (A, B) and (B, A) count as different outcomes.

Binomial coefficients form the rows of Pascal's Triangle and the coefficients of the binomial expansion (x + y)ⁿ, and they underlie the binomial probability distribution used to calculate the odds of exactly k successes in n independent trials. Statisticians, probability students, poker and lottery-odds calculators, and combinatorics researchers rely on C(n, k) constantly — anytime a problem asks “how many ways can you choose...” without regard to order, this is the formula being applied.

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Comments (1)

  • Editorial team

    The tool gives you the count of ways, which is only half of a probability question. To get the chance of exactly k successes in n independent trials, multiply C(n, k) by p to the power k and by (1 minus p) to the power n minus k. For 3 heads in 10 fair coin flips that is 120 × (1/1024), about 11.7%. For a pick-6-of-49 draw, C(49, 6) shows 13,983,816 combinations. What problem are you counting?

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