Answer
The number of different ways to arrange the program for the segment of a radio show is $8648640$.
Work Step by Step
A disc jockey is choosing 7 songs from a group of 13 songs. The order in which the songs are chosen matters because each song is different. Therefore, we need to find the number of permutations of 13 things taken 7 at a time.
By using the formula,
${}_{n}{{P}_{r}}=\frac{n!}{\left( n-r \right)!}$
Where $ n=13,r=7$
${}_{13}{{P}_{7}}=\frac{13!}{\left( 13-7 \right)!}=\frac{13!}{6!}$
$\begin{align}
& =\frac{13\times 12\times 11\times 10\times 9\times 8\times 7\times 6!}{6!} \\
& =13\times 12\times 11\times 10\times 9\times 8\times 7 \\
& =8648640 \\
\end{align}$
