What is a Binomial Coefficient?


A binomial coefficient tells us how many ways we can choose k things out of n total things.

A binomial coefficient is written as follows:

Binomial coefficient

where:

  • n: The total number of things (n ≥ 0)
  • k: The size of the subset (k ≤ n)
  • !: A symbol that means factorial

We typically pronounce this as “n choose k” and sometimes write it as nCk.

Example: Calculate the Binomial Coefficient

For example, suppose we have three widgets – call them “A”, “B”, “C”, and “D” – and we’d like to know how many different ways we can choose 2 of these widgets.

We can use the binomial coefficient formula to find this out:

Binomial coefficient example

There are 6 ways to choose 2 things out of 4 total things. We can even list out each combination if we’d like:

  • A, B
  • A, C
  • A, D
  • B, C
  • B, D
  • C, D

Note: We can use this calculator to confirm that there are 6 ways to choose k = 2 things out of n = 4 total things.

The Binomial Coefficient in Practice

In practice, the binomial coefficient shows up in the formula for the Binomial distribution, which tells us the probability of obtaining k success in n trials.

If a random variable X follows a binomial distribution, then the probability that X = k successes can be found by the following formula:

P(X=k) = nCk * pk * (1-p)n-k

where:

  • n: number of trials
  • k: number of successes
  • p: probability of success on a given trial
  • nCkthe number of ways to obtain k successes in n trials

The binomial coefficient appears in the beginning of this formula and it tells us how many different ways k successes could occur in n total trials.

P(X=k) = nCk * pk * (1-p)n-k

The way we use this formula is straightforward. Suppose we flip a coin 3 times and we’d like to know the probability that it lands on heads 2 times:

P(X=2) 3C2 * .52 * (1-.5)3-2 = 3 * .25 * (.5)1 = 0.375

The probability that it lands on heads 2 times is 0.375.

Additional Resources

An Introduction to the Binomial Distribution
An Introduction to Binomial Experiments
How to Calculate Binomial Probabilities on a TI-84 Calculator

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