Converting Permutations to Combinations: Simple Stats in Practise In a popular text book there is a question that has bothered me that I am sure is very simple for others and I'm just missing something.....
So image $100$ songs and we have $10$ as Beatles songs. We want to pick $5$ songs where only the last one we pick is a Beatles song. The probability of that happening expressed as a permutation is ...
$\dfrac{ (90 * 89 * 88 * 87) * 10 }{ 100 * 99 * 98 * 97 * 96 } = \dfrac{ P(4,90) * 10 }{ P(5,100) }$.
This makes sense to me. But then they translate it to a combination to show us another way of thinking about it.
$\dfrac{C(95,9)}{C(100,10)}$
This part is hard for me to grasp. Initially I thought that it describes the chance that $9$ would appear in $90$ if there are $10$ in $100$. That got clarified by a friend of mine when he explained it actually is a sort of logical implication because the last Beatles song will never be in the 95 set so the probability of $9$ in $95$ should somewhat match the probability of $1$ in the last $5$.
So here is my REAL question... How is it that the combination also expresses the last song being the beatles song as opposed to it being one of the first four.
If there is any clarification that is needed please let me know!! I would really love to answer this question. I'm doing this out of deep interest for the game of Mathematics.
 A: It appears from your calculations that where you say "the last one we pick is a Beatles song" you intended to imply that the other four songs that are picked are not Beatles songs.
Your idea "the chance that $9$ would appear in $90$ if there are $10$ in $100$" doesn't fit because the upper index in the numerator is $95$, not $90$. The first calculation focusses on the choices for the selected songs, of which there are $5$. The second calculation focusses on the choices for the Beatles songs, of which there are $10$.
Imagine that you choose not just $5$ records to play but a playlist that determines the order in which you'll play all $100$ records. The $5$ records actually selected are the first $5$ on this list of $100$. Now consider the slots at which the Beatles songs appear. In total, there are $\binom{100}{10}$ ways to choose these slots. But if the first four shouldn't be Beatles songs and the fifth one should, that leaves $95$ slots from which to choose $9$ for the remaining Beatles songs, yielding $\binom{95}9$ choices.
