EXAMPLE 14 Playing Songs

Chapter 7, Example 4 (page 299), involved choosing a random sample of four different songs from a digital media player with a playlist of 27 different songs from Professor Lesser’s The Beatles One album. Now we ask: How many 4-song samples are possible from a collection of 27 songs? Like DNA sequences in Example 13, order matters here. (Performers and DJs know that the same four songs can feel quite different when the songs are played in a different order.) Unlike DNA sequences, listing the same item more than once is not allowed.

Any of the 27 songs can be chosen to be played first, but only the remaining 26 songs are available to be listed as the second song, so that there are choices for the first two songs. Any of these choices leaves 25 songs for the third position and 24 for the fourth position. Surprisingly, the number of playlists of four different songs chosen from a list of 27 songs is almost half a million!

Now, suppose Professor Lesser’s favorite song is “Let it Be.” What is the probability that a randomly chosen playlist of four different songs will include this song? To answer the question, let event be the playlists that include “Let it Be.” In order to apply the procedure for finding probabilities of equally likely outcomes, we need to count the number of outcomes in A. “Let it Be” could be the first, second, third, or fourth song on a playlist. If it is the first song, then ways to complete the playlist by choosing three songs in order from the remaining 26 songs. The same is true if “Let it Be” is the second, third, or fourth song. Therefore, the number of outcomes in is and . Thus, “Let it Be” will be included roughly 15% of the time in randomly selected playlists of four different songs.

This scenario of choosing an ordered subset of songs from a playlist of songs is called a permutation.