Answer
The number of ways to award the prize to three people out of 50 people (if the first prize is $\$1000$, second prize is $\$500$, and third prize is $\$100$) is $117,600$.
Work Step by Step
We know that the number of ways in which r number of things are arranged from n number of things is obtained by:
${}_{n}{{P}_{r}}=\frac{n!}{\left( n-r \right)!}$
So, from the information,
$\begin{align}
& n=50 \\
& r=3 \\
\end{align}$
Then, we have to find the number of permutations of 50 things taken 3 at a time:
$\begin{align}
& {}_{50}{{P}_{3}}=\frac{50!}{\left( 50-3 \right)!} \\
& =\frac{50!}{47!} \\
& =\frac{50\times 49\times 48\times 47!}{47!} \\
& =117600
\end{align}$
Thus, the number of ways is $117,600$.
