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## Tunes for Cruising

Probability puzzles require you to weigh all the possibilities and pick the most likely outcome.

 Puzzle ID: #14898 Fun: (2.5) Difficulty: (2.67) Category: Probability Submitted By: snaps

I recently burnt a CD with a selection of songs from my favourite rock bands to listen to in my car. The order of tracks, along with the bands that sing them, as burnt onto the CD are listed below (note: these songs and bands may sound similar to some more well-known songs and bands but that's purely coincidental).

1. Sourest Thing - V3
2. I Still Haven't Heard What I'm Listening For - V3
3. Without and With Me - V3
4. Son - Pearl Honey
5. Spin the White Square - Pearl Honey
6. Better Woman - Pearl Honey
7. Last Hug - Pearl Honey
9. Off He Comes - Pearl Honey
10. Finding Your Religion - M.E.R.
11. Nobody Hurts - M.E.R.

When driving along listening to this fine collection of tunes, I usually put my CD player on random (ie. the order that the tracks are played does not follow any set pattern). If my CD player plays through all 11 tracks before repeating any of them, what is the probability that no two songs from the same band will be played one after the other?

The probability of playing the tracks in a random order without any two songs from the same band being played one after the other is 1/462.

As there are six Pearl Honey songs they must be played in the odd-numbered positions. The V3 and M.E.R. songs will be played in the even-numbered positions. One such example is shown below:

1. Spin the White Square - Pearl Honey
2. I Still Haven't Heard What I'm Listening For - V3
3. Off He Comes - Pearl Honey
4. Finding Your Religion - M.E.R.
5. Better Woman - Pearl Honey
6. Without and With Me - V3
7. Last Hug - Pearl Honey
8. Nobody Hurts - M.E.R.
10. Sourest Thing - V3
11. Son - Pearl Honey

There are six possible Pearl Honey songs that could be played first.
There are five possible V3 or M.E.R. songs that could be played second.
For the third song, there are now only five possible Pearl Honey songs that could be played.
For the fourth song, there are now only four possible V3 or M.E.R. songs that could be played.
And so on and so forth...

Following this pattern we have that there are 6*5*5*4*4*3*3*2*2*1*1 = 86,400 playlists where no two songs by the same band are played one after the other.

All up there are 11*10*9*8*7*6*5*4*3*2*1 = 39,916,800 different playlists in which the 11 tracks could be played.

So the probability that all 11 tracks will be played without any two songs from the same band being played one after the other is 86,400/39,916,800 which can be simplified to 1/462.

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