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:

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 _{n}C_{k}.

**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:

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) = _{n}C_{k} * p^{k} * (1-p)^{n-k}**

where:

**n:**number of trials**k:**number of successes**p:**probability of success on a given trialthe number of ways to obtain_{n}C_{k}:*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) = _{n}C_{k} * p^{k} * (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) **= _{3}C_{2} * .5^{2} * (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