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.
The binomial coefficient is denoted byand it is read as " choose " or " over ".
It is defined as follows:where the exclamation mark denotes a factorial.
Reminder: remember that the factorial of a natural number is equal to the product of all natural numbers less than or equal to :and that, by convention, .
In combinatorics, the binomial coefficient indicates the number of possible combinations of objects from .
Example The number of possible ways to choose 2 objects from a set of 5 objects is equal to
When we deal with combinations, we need to keep in mind that:
the order in which the objects are selected does not matter;
each object can be selected only once.
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 isIf 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
In algebra, the binomial coefficient is used to expand powers of binomials. According to the binomial theorem,
Example The third power of a binomial can be expanded as follows:
If we replace with in the formula above, we can see that is the coefficient of in the expansion of . This is often presented as an alternative definition of the binomial coefficient.
The binomial coefficient is used in probability and statistics, most often in the binomial distribution, which is used to model the number of positive outcomes obtained by repeating times an experiment that can have only two outcomes (success and failure).
More details can be found in the lecture entitled Combinations, where:
we explain why combinations can be counted using binomial coefficients;
we report some useful recursive formulae;
we introduce combinations with repetition and multiset coefficients;
we make more examples;
we propose some solved exercises.
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Please cite as:
Taboga, Marco (2021). "Binomial coefficient", Lectures on probability theory and mathematical statistics. Kindle Direct Publishing. Online appendix. https://www.statlect.com/glossary/binomial-coefficient.
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