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
by![[eq1]](/images/binomial-coefficient__1.png){kind=small}
and
it is read as
"
choose
"
or
"
over
".
It is defined as
follows:![[eq2]](/images/binomial-coefficient__6.png)
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
:![[eq3]](/images/binomial-coefficient__9.png){kind=small}
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![[eq4]](/images/binomial-coefficient__13.png)
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
is![[eq5]](/images/binomial-coefficient__15.png)
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]](/images/binomial-coefficient__16.png){kind=small}
In algebra, the binomial coefficient is used to expand powers of binomials.
According to the binomial
theorem,![[eq7]](/images/binomial-coefficient__17.png)
Example
The third power of a binomial can be expanded as
follows:![[eq8]](/images/binomial-coefficient__18.png)
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.
Previous entry: Alternative hypothesis
Next entry: Conditional probability density function
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.
Most of the learning materials found on this website are now available in a traditional textbook format.
