Question

In the following exercises, evaluate.

(a) $\left(\begin{array}{l}3\\ 1\end{array}\right)$ (b) $\left(\begin{array}{l}9\\ 9\end{array}\right)$ (c) $\left(\begin{array}{l}7\\ 0\end{array}\right)$ (d)$\left(\begin{array}{l}5\\ 3\end{array}\right)$

Short answer

(a)$3$

(b)$1$

(c)$1$

(d)$10$

Step-by-step solution

Part (a) Step 1 Given information 

Evaluate$\left(\begin{array}{l}3\\ 1\end{array}\right)$

Part (a) Step 2 Solution 

$\begin{array}{rcl}\left(\begin{array}{l}3\\ 1\end{array}\right)& =& \frac{3!}{(3-1)!·1!}\\ & =& 3\end{array}$

Part (b) Step 1 Given information 

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

Part (b) Step 2 Solution 

$\begin{array}{rcl}\left(\begin{array}{l}9\\ 9\end{array}\right)& =& \frac{9!}{(9-9)!·9!}\\ & =& 1\end{array}$

Part (c) Step 1 Given information 

Evaluate$\left(\begin{array}{l}7\\ 0\end{array}\right)$

Part (c) Step 2 Solution 

$\begin{array}{rcl}\left(\begin{array}{l}7\\ 0\end{array}\right)& =& \frac{7!}{(7-0)!·0!}\\ & =& 1\end{array}$

Part (d) Step 1 Given information 

Evaluate$\left(\begin{array}{l}5\\ 3\end{array}\right)$

Part (d) Step 2 Solution 

$\begin{array}{rcl}\left(\begin{array}{l}5\\ 3\end{array}\right)& =& \frac{5!}{(5-3)!·3!}\\ & =& 10\end{array}$