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)
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 the 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)
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)
In algebra, it is used to expand powers of binomials. According to the
binomial
theorem,![[eq5]](/images/binomial_coefficient__14.png)
Example
The third power of a binomial can be expanded as
follows:![[eq6]](/images/binomial_coefficient__15.png)
More details can be found in the lecture entitled Combinations, where we explain why combinations can be counted using binomial coefficients, and where we also report some useful recursive formulae that can be used to calculate binomial coefficients.
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