Question

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?

Short answer

The number of ways in which eight songs can be arranged to form eight different playlists is 40320.

The probability that the songs are arranged in the alphabetical order of their title is$\frac{1}{40320}$.

Step-by-step solution

Given information

There are eight songs to make eight different playlists.

Define factorial

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:

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

Compute the number of arrangements

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.

Compute the probability of picking a list in the alphabetical order of song title

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.