In Exercises 5–36, express all probabilities as fractions.

Party Mix DJ Marty T is hosting a party tonight and has chosen 8 songs for his final set (including “Daydream Believer” by the Monkees). How many different 8-song playlists are possible (song order matters)? If the 8 songs are randomly selected, what is the probability they are in alphabetical order by song title?

There are eight songs to make eight different playlists.

The number of different ways in which r units can be arranged in r number of places is $r!$.

The factorial can be elaborated as follows:

$r!=r\times \left(r-1\right)\times \left(r-2\right)\times ...\times 1$

There is a total of eight songs.

The number of ways in which eight songs can be arranged to form a playlist is

$$\begin{gathered} 8!=8\times 7\times 6\times 5\times 4\times 3\times 2\times 1 \\ =40320 \end{gathered}$$

Therefore, the number of possible eight-song playlists is 40320.

The total number of playlists possible is 40320.

The number of ways of forming a playlist with songs arranged according to the alphabetical order of their title is 1.

The probability thatthe songs are arranged in the alphabetical order of their title is

$$\begin{gathered} P\left(\text{songs}\text{arranged}\text{in}\text{alphabetical}\text{order}\right)=\frac{\text{Number}\text{of}\text{favourable}\text{ways}}{\text{Total}\text{number}\text{of}\text{ways}} \\ =\frac{1}{40320} \end{gathered}$$

Therefore,the probability that the songs are arranged in the alphabetical order of their title is.