Question

How do you find the binomial coefficient of \left( {\begin{array}{*{20}{l}}{100}\\\\{98}\end{array}} \right)?

Answer 1

Here we need to find the binomial coefficient of the given expression. We will first compare the general coefficient of the binomial with the given coefficient to find the values of the terms. Then we will substitute these values in the formula of combination to find the required value.
Formula used: The formula of combination is ${}^n{C_r} = \dfrac{{n!}}{{\left( {n - r} \right)! \times r!}}$.

Complete step by step solution:
Here we need to find the binomial coefficient of the given expression.
Here, we have
\left( {\begin{array}{*{20}{l}}{100}\\\\{98}\end{array}} \right)…………… $\left( 1 \right)$
We know that the coefficient of binomial ${C_r}$ of ${x^r}$ in ${\left( {1 + x} \right)^n}$ are denoted by \left( {\begin{array}{*{20}{l}}n\\\r\end{array}} \right)………… $\left( 2 \right)$
On comparing equation $\left( 1 \right)$ with equation $\left( 2 \right)$, we get
$n = 100$ and $r = 98$
Now, we will find the value of coefficients using the formula.
Therefore, the value of coefficient $= {}^{100}{C_{98}}$
We know the property of combination that ${}^n{C_r} = {}^n{C_{n - r}}$.
Now, using this property of combination, we get
$\Rightarrow$ The value of coefficient $= {}^{100}{C_{100 - 98}}$
On subtracting the numbers, we get
$\Rightarrow$ The value of coefficient $= {}^{100}{C_2}$
Using this formula of combination here, we get
$\Rightarrow$ The value of coefficient $= \dfrac{{100!}}{{\left( {100 - 2} \right)! \times 2!}}$
On subtracting the numbers inside the bracket, we get
$\Rightarrow$ The value of coefficient $= \dfrac{{100!}}{{98! \times 2!}}$
Now, we will find the value of factorials here.
$\Rightarrow$ The value of coefficient $= \dfrac{{100 \times 99 \times 98!}}{{98! \times 2 \times 1}}$
On further simplification, we get
$\Rightarrow$ The value of coefficient $= 50 \times 99$
On multiplying the numbers, we get
$\Rightarrow$ The value of coefficient $= 4950$

Hence, the required value of the binomial coefficient is equal to 4950.

Note:
Here we have obtained the value of binomial coefficients. The binomial coefficient is denoted by \left( {\begin{array}{*{20}{l}}n\\\r\end{array}} \right) and it is defined as the number of ways to select $r$ unordered outcomes from $n$ possibilities, which is also known as a combination or combinatorial number. The symbols \left( {\begin{array}{*{20}{l}}n\\\r\end{array}} \right) and ${}^n{C_r}$ are generally used to represent or to denote a binomial coefficient.