Suppose you are a DJ and you are given a list of 15 pop song

Suppose you are a DJ and you are given a list of 15 pop songs and 5 classic rock songs that you are to play consecutively.

a. How many ways are there to play the songs if each is distinct and you don’t care how many of each are played consecutively? Explain your response.

b. Suppose that your tolerance for pop music is low, so you would like to create a list that minimizes the number of consecutive pop songs. How many consecutive pop songs must you play? Explain your response.

c. Give an example of a playlist (P for pop song, R for classic rock) that meets the condition you determined in part b.

d. I have a friend that thinks all pop music sounds the same. In order words, according to this person, playing fifteen pop songs is the same as playing one of them. How many ways are there to play the songs in the pop songs are considered identical but the classic rock are considered distinct? Explain your response.

Solution

a) There are total 20 songs and there is no restriction on how DJ plays them.

Thus, number of ways = 20!

b) If pop consecutive have to minimized, DJ should start with Pop song and then classic....and so on

P R P R P R P R P R .......10 consecutive Pops now

So, minimum 10 consecutive pops have to be played.

c) Playlist:

P R P R P R P R P R .......10 consecutive Pops now