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Binomial coefficient

by , PhD

In combinatorics, the binomial coefficient is used to denote the number of possible ways to choose a subset of objects of a given numerosity from a larger set.

It is so called because it can be used to write the coefficients of the expansion of a power of a binomial.

Table of Contents


The binomial coefficient is denoted by[eq1]and it is read as "n choose k" or "n over k".


It is defined as follows:[eq2]where the exclamation mark denotes a factorial.

Reminder: remember that the factorial of a natural number n is equal to the product of all natural numbers less than or equal to n:[eq3]and that, by convention, the factorial of zero is one.

Usage in combinatorics

In combinatorics, the binomial coefficient indicates the number of possible combinations of k objects from n.

Example The number of possible ways to choose 2 objects from a set of 5 objects is equal to[eq4]

When we deal with combinations, we need to keep in mind that:

If the latter requirement is violated, then we are dealing with combinations with repetition. In that case, we cannot use binomial coefficients, but we need to use multiset coefficients.

Example There is a basket of fruits containing pears, bananas, oranges and apples. The choice of two different fruits from the basket is a combination, and the number of possible choices is[eq5]If you choose first an apple and then an orange, that is the same thing as picking first an orange and then an apple. These two choices are counted as a single combination. If we allow for the possibility of selecting two fruits of the same kind (e.g., two bananas), then we are dealing with combinations with repetition and the number of possible selections is given by the multiset coefficient[eq6]

Depending on whether we allow or not for repetitions, the number of combinations of two fruits chosen from four is either six or ten.

Usage in algebra

In algebra, the binomial coefficient is used to expand powers of binomials. According to the binomial theorem,[eq7]

Example The third power of a binomial can be expanded as follows:[eq8]

If we replace $b$ with 1 in the formula above, we can see that $inom{n}{k}$ is the coefficient of $a^{k}$ in the expansion of [eq9]. This is often presented as an alternative definition of the binomial coefficient.

Usage in probability and statistics

The binomial coefficient is used in probability and statistics, most often in the binomial distribution, which is used to model the number k of positive outcomes obtained by repeating n times an experiment that can have only two outcomes (success and failure).

More details

More details can be found in the lecture entitled Combinations, where:

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How to cite

Please cite as:

Taboga, Marco (2021). "Binomial coefficient", Lectures on probability theory and mathematical statistics. Kindle Direct Publishing. Online appendix.

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