$\displaystyle \binom{n}{r}$ $\,=\,$ $\dfrac{n!}{r!(n-r)!}$

A numeral coefficient of a factor in variable form in a term of binomial theorem’s expansion is called a binomial coefficient.

In the expansion of Binomial Theorem, each term is formed by the product of a quantity in numeral form and a quantity in literal form. Due to their multiplication, the numeral quantity is usually called the coefficient of the quantity in literal form and is also simply called the numeral coefficient of the quantity in literal form. The numeral coefficient of the factor in variable form is appeared in each term of the binomial theorem’s expansion. Hence, it is called the binomial coefficient.

Let $n$ and $r$ be two constants and they both are positive integers, which means $n \ge 0$ and $k \ge 0$. Similarly, the value of $n$ is always greater than or equal to $k$. It means $n \ge k$. In this case, a binomial coefficient is written in the following mathematical form.

$\displaystyle \binom{n}{r}$

The symbolical representation of a binomial coefficient is read as $n$ choose $k$.

$\displaystyle \binom{5}{2}$

The binomial coefficient is $5$ choose $2$.

Let us consider a set $\big\{a,\, b,\, c,\, d,\, e\big\}$. The total members in this set are $5$. Now, choose a subset of $2$ elements in possible ways from the set of $5$ members to find the value of binomial coefficient $5$ choose $2$.

$(1).\,\,$ $\big\{a,\, b\big\}$

$(2).\,\,$ $\big\{a,\, c\big\}$

$(3).\,\,$ $\big\{a,\, d\big\}$

$(4).\,\,$ $\big\{a,\, e\big\}$

$(5).\,\,$ $\big\{b,\, c\big\}$

$(6).\,\,$ $\big\{b,\, d\big\}$

$(7).\,\,$ $\big\{b,\, e\big\}$

$(8).\,\,$ $\big\{c,\, d\big\}$

$(9).\,\,$ $\big\{c,\, e\big\}$

$(10).\,\,$ $\big\{d,\, e\big\}$

It is cleared that the possible ways to choose a subset of $2$ elements from a set of $5$ members is $10$. Therefore, the value of binomial coefficient $5$ choose $2$ is $10$.

$\therefore\,\,\,$ $\displaystyle \binom{5}{2} \,=\, 10$

The binomial coefficient is also expressed in the following forms. So, you can denote a binomial coefficient in your convenient form.

$(1).\,\,$ $^nCr$

$(2).\,\,$ $nCr$

$(3).\,\,$ $C\,n, r$

$(4).\,\,$ $C(n, r)$

$(5).\,\,$ $C_{n}^{r}$

$(6).\,\,$ $C_{r}^{n}$

The binomial coefficient can be evaluated mathematically by the following formula.

$\displaystyle \binom{n}{r}$ $\,=\,$ $\dfrac{n!}{r!(n-r)!}$

Let’s find the binomial coefficient for the above example by this formula.

$\displaystyle \binom{5}{2}$ $\,=\,$ $\dfrac{5!}{2!(5-2)!}$

$\,\,\,=\,$ $\dfrac{5!}{2!(3)!}$

$\,\,\,=\,$ $\dfrac{5!}{2! \times 3!}$

$\,\,\,=\,$ $\dfrac{5 \times 4 \times 3 \times 2 \times 1}{2 \times 1 \times 3 \times 2 \times 1}$

$\,\,\,=\,$ $\dfrac{5 \times \cancel{4} \times \cancel{3 \times 2 \times 1}}{\cancel{2} \times 1 \times \cancel{3 \times 2 \times 1}}$

$\,\,\,=\,$ $\dfrac{5 \times 2}{1 \times 1}$

$\,\,\,=\,$ $\dfrac{10}{1}$

$\,\,\,=\,$ $10$

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