Simone is a radio producer. Her role is to create a play list that is 8 songs long. These 8 songs must all be from the top 8 voted songs from the

Question

listeners. No song may be repeated. There are 2 Britney Spears songs and 2 Akon songs in the top 8. How many ways can Simone create the play list if two consecutive songs cannot be by the same singer?

Answer is 23 040 but I have no idea how to do it.

Answer 1

We can solve this problem using the principle of Inclusion-Exclusion.

First, consider the total number of ways to arrange the 8 songs without any restrictions. Since there are 2 Britney Spears songs and 2 Akon songs, there are 8!/(2!2!) ways to arrange the songs (dividing by 2! twice because there are 2 repeated songs for both artists). That is equal to 10080.

Now, we need to subtract the number of ways that at least one pair of the same artist's songs are consecutively played. We can perform "block counting" method for this situation: imagine the two Britney Spears songs as a single block. Then, we will have 7 "items" to arrange: the block of Britney Spears songs, the 2 Akon songs, and the remaining 4 songs. There are 7!/2! ways to arrange these (dividing by 2! again because of the repeated Akon songs). That is equal to 2520.

However, we also have to consider the cases when two Akon songs are together or both pairs of songs are consecutively played. Applying the same block counting technique, we also have 2520 arrangements for two Akon songs together and 6! arrangements for all four Britney Spears and Akon songs being together in blocks.

Summing up these overlapping cases, and we get 2520 + 2520 + 720 = 5760.

By the principle of Inclusion-Exclusion, we need to subtract the overlapping cases from the total number of ways: 10080 - 5760 = 4320.

Now, we have to consider the cases when three consecutive songs are of the same artists (two from one artist and one from the other). We only have two cases here, one is 2 Britney Spears songs followed by 1 Akon song and another one is 2 Akon songs followed by 1 Britney Spears song.

In the first case, treat the 2 Britney Spears songs as a block and the 2 Akon songs as a block making it the third position. Then, we have 6!/(2!1!) (dividing by 2! due to the 1 repeated Akon song) ways to arrange the items. That is equal to 360.

Similarly, in the second case, we have 6!/(1!2!) (dividing by 2! because there is 1 repeated Britney Spears song) ways. That is also equal to 360.

So, adding these back to our previous result (Inclusion-Exclusion):

4320 + 360 + 360 = 5040.

Finally, consider the case where four consecutive songs include both the Britney Spears songs and both Akon songs in any order. We have 2! (BS-BS-AK-AK or AK-AK-BS-BS) ways of arranging the 4 consecutive songs and 5! ways of arranging the whole sequence with these 4 consecutive songs.

Thus, there are 2! * 5! = 240 ways for this case.

So, subtracting this case (Inclusion-Exclusion):

5040 - 240 = 4800.

The final answer is 4800 + 240 = 5040 different playlists, which is incorrect. The correct answer should be 23,040 as provided in the question, and our solution seems to have an error.