Question

In the following exercises, evaluate.

(a) $\left(\begin{array}{l}4\\ 1\end{array}\right)$ (b) $\left(\begin{array}{l}5\\ 5\end{array}\right)$ (c) $\left(\begin{array}{l}8\\ 0\end{array}\right)$ (d)$\left(\begin{array}{l}11\\ 9\end{array}\right)$

Short answer

(a)$4$

(b)$1$

(c) $1$

(d)$55$

Step-by-step solution

Part (a) Step 1 Given information 

Evaluate $\left(\begin{array}{l}4\\ 1\end{array}\right)$

Part (a) Step 2 Solution 

$\begin{array}{rcl}\left(\begin{array}{l}4\\ 1\end{array}\right)& =& \frac{4!}{(4-1)!·1!}\\ & =& 4\end{array}$

Part (b) Step 1 Given information 

Evaluate$\left(\begin{array}{l}5\\ 5\end{array}\right)$

Part (b) Step 2 Solution 

$\begin{array}{rcl}\left(\begin{array}{l}5\\ 5\end{array}\right)& =& \frac{5!}{(5-5)!·5!}\\ & =& 1\end{array}$

Part (c) Step 1 Given information 

Evaluate$\left(\begin{array}{l}8\\ 0\end{array}\right)$

Part (c) Step 2 Solution 

$\begin{array}{rcl}\left(\begin{array}{l}8\\ 0\end{array}\right)& =& \frac{8!}{(8-0)!·0!}\\ & =& 1\end{array}$

Part (d) Step 1 Given information 

Evaluate$\left(\begin{array}{l}11\\ 9\end{array}\right)$

Part (d) Step 2 Solution 

$\begin{array}{rcl}\left(\begin{array}{l}11\\ 9\end{array}\right)& =& \frac{11!}{(11-9)!·9!}\\ & =& 55\end{array}$