My friends and I are discussing the randomness of shuffling a playlist.
Lets say I have a playlist of $700$ songs, $30$ of which are by the same artist (its Pendulum, don't ask).
My understanding is that the number of possible shuffled playlists is equal to $700!$ (factorial, not enthusiasm). Assuming that the songs are not repeated after they are selected, that means that the probability of selecting any given sequence of songs is $\frac{1}{700!}$
How would I calculate the probability that all $30$ pendulum songs are played back to back? My gut says that this would be equal to $\frac{1}{700} \cdot \frac{1}{699}\cdot \frac{1}{698} \cdots \frac{1}{670}$. Is this instinct correct and if so, is there a more concise formula?
I used to know this stuff but haven't studies maths since I was $18$, any suggestions would be appreciated!
Without effort questions goes out of community standrad. Still I am answering it.
Let, $30$ pendulum songs as $1$ unit song.
Now, $670+1=671$ songs can be arranged in $671!$ ways and $30$ songs can be arranged in $30!$ ways.
So, there is $671!\times 30!$ ways out of $700!$ possible ways to listen all pendulum songs in a row.
Threfore, Probability to listen all pendulum songs in a row$=\frac{671!\times30!}{700!}=(30!)\times\frac{1}{700} \cdot \frac{1}{699}\cdot \frac{1}{698} \cdots \frac{1}{670}$