Math July 13, 2026 · 8 Min Read

Binomial Coefficient Calculator – Guide & Formulas

Calculate binomial coefficients C(n,k) = n choose k instantly. See the formula, Pascal's triangle connection, and step-by-step factorial expansion.

Calculate the binomial coefficient C(n,k) — also known as "n choose k" — with our free calculator. Get the exact value, formula expansion, and Pascal's triangle position.

Key Takeaway

Use the free Binomial Coefficient Calculator to calculate binomial coefficients c(n,k) = n choose k instantly. see the formula, pascal's triangle connection, and step-by-step factorial expansion. Get instant results with step-by-step explanations.

How to Use the Binomial Coefficient Calculator

  1. Enter n: the total number of items (non-negative integer).
  2. Enter k: the number of items to choose (integer, 0 <= k <= n).
  3. Review the binomial coefficient value C(n,k).
  4. See the factorial expansion and step-by-step simplification.

The Formula

C(n,k) = n! / (k! × (n-k)!), where n! = n × (n-1) × ... × 1. This counts the number of ways to choose k items from n items without regard to order.

Variable Definitions

  • C(n,k): Binomial coefficient — number of ways to choose k from n
  • n!: Factorial of n: product of all positive integers up to n
  • n: Total number of items
  • k: Number of items to choose

Computing C(8, 3)

Find the number of ways to choose 3 items from 8.

  1. Step 1: Identify: n = 8, k = 3.
  2. Step 2: Apply formula: C(8,3) = 8! / (3! × 5!).
  3. Step 3: Expand: 8! = 40320, 3! = 6, 5! = 120.
  4. Step 4: Calculate: 40320 / (6 × 120) = 40320 / 720 = 56.
  5. Step 5: There are 56 ways to choose 3 items from 8.

Frequently Asked Questions

What does "n choose k" mean?

"n choose k" counts the number of ways to select k items from a set of n distinct items, where order does not matter. For example, choosing 2 toppings from 5 gives C(5,2) = 10 different combinations.

What is the relationship to Pascal's Triangle?

Each entry in Pascal's Triangle is a binomial coefficient. Row n, position k contains C(n,k). For example, row 5 is 1, 5, 10, 10, 5, 1 — these are C(5,0) through C(5,5).

What if k = 0 or k = n?

C(n,0) = 1 and C(n,n) = 1. There is exactly one way to choose nothing or to choose everything.

What if k > n?

C(n,k) = 0 when k > n, because you cannot choose more items than are available. The calculator returns 0 for this case.

How is this different from permutations?

Permutations P(n,k) count arrangements where order matters: P(n,k) = n!/(n-k)!. Binomial coefficients C(n,k) count combinations where order does not matter: C(n,k) = P(n,k)/k!.