Answer
$\displaystyle \left(\begin{array}{l} 5\\ 0 \end{array}\right)-\left(\begin{array}{l} 5\\ 1 \end{array}\right)+\left(\begin{array}{l} 5\\ 2 \end{array}\right)-\left(\begin{array}{l} 5\\ 3 \end{array}\right)+\left(\begin{array}{l} 5\\ 4 \end{array}\right)-\left(\begin{array}{l} 5\\ 5 \end{array}\right)=0$
Work Step by Step
We evaluate:
$\displaystyle \left(\begin{array}{l} 5\\ 0 \end{array}\right)-\left(\begin{array}{l} 5\\ 1 \end{array}\right)+\left(\begin{array}{l} 5\\ 2 \end{array}\right)-\left(\begin{array}{l} 5\\ 3 \end{array}\right)+\left(\begin{array}{l} 5\\ 4 \end{array}\right)-\left(\begin{array}{l} 5\\ 5 \end{array}\right)=\frac{5!}{0!5!}-\frac{5!}{1!4!}+\frac{5!}{2!3!}-\frac{5!}{3!2!}+\frac{5!}{4!1!}-\frac{5!}{5!0!}=0$
Because all the terms cancel.