Chapter 13.2, Problem 2AYU

To determine

To calculate: The binomial coefficient of $\left(\begin{array}{l}6\\ 4\end{array}\right)$ equals from the given choices.

$\frac{6!}{4!}$(b) $\frac{6!}{4!\cdot 2!}$(c) $\frac{6!}{2!}$(d) $\frac{\left(6-4\right)!}{2!}$

Answer to Problem 2AYU

Solution:

The binomial coefficient of $\left(\begin{array}{l}6\\ 4\end{array}\right)$ equals $\frac{6!}{4!\cdot 2!}$

Explanation of Solution

Given information:

The binomial is $\left(\begin{array}{l}6\\ 4\end{array}\right)$ and the choices are

$\frac{6!}{4!}$(b) $\frac{6!}{4!\cdot 2!}$(c) $\frac{6!}{2!}$(d) $\frac{\left(6-4\right)!}{2!}$

Formula used:

The formula for $\left(\begin{array}{l}n\\ r\end{array}\right)$ is $\left(\begin{array}{l}n\\ r\end{array}\right)=\frac{n!}{\left(n-r\right)!\cdot r!}$

Calculation:

Comparing $\left(\begin{array}{l}n\\ r\end{array}\right)$ with $\left(\begin{array}{l}6\\ 4\end{array}\right)$, $n=6$ and $r=4$,

$\left(\begin{array}{l}n\\ r\end{array}\right)=\left(\begin{array}{l}6\\ r\end{array}\right)=\frac{6!}{\left(6-4\right)!\cdot 4!}$

$\Rightarrow \left(\begin{array}{l}6\\ r\end{array}\right)=\frac{6!}{2!\cdot 4!}$

So, the binomial $\left(\begin{array}{l}6\\ 4\end{array}\right)$ equals $\frac{6!}{4!\cdot 2!}$.