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I am having difficulty with one minuscule detail of the permutation formula:


I understand that if we proceed with an $r$-permutation, then we have $r$ amount of slots, where the first slot has a total of $n$ ways of being occupied by a single object, from a set of n objects; similarly, for the second slot, we have one less object from the total amount (because one has already been selected for the first slot), meaning we $n-1$ amount of ways to select a single object for it. I also understand that as you go down the line of slots, there are less and less objects available to select for a particular slot that is relatively far from the first slot. My question is, why does the amount of decrease such that the last slot has a total of $n-r+1$ ways to select an object for that slot?

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up vote 2 down vote accepted

The permutation formula is $$P(n,r)=\frac{n!}{(n-r)!}=\frac{n(n-1)\cdots(n-r+1)(n-r)(n-r-1)\cdots2\cdot1}{(n-r)(n-r-1)\cdots2\cdot1}$$

This is why the last slot is $(n-r+1)$. Think of it in concrete terms. I have 8 objects, and I want to take 4 of those objects and see how many permutations I can generate with those 4 objects. So $n=8$ and $r=4$. Thus, the number of ways I can do it is $$8\cdot7\cdot6\cdot5$$ What is 5? It is $8-4+1$...

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Why do you write things like $8*7*6*5$ instead of $8\cdot7\cdot6\cdot5$ or $8\times7\times6\times5$? The whole point of using an asterisk is that one is limited to the characters on the keyboard, so that things like $\times$ are not available, and one want to avoid using $x$ instead of $\times$ because the letter $x$ should be available for other uses. And the whole point of $\TeX$ is to make things like $8\cdot7$ and $8\times7$ available. Using an asterisk for ordinary multiplication in $\TeX$ amounts to eating mashed potatoes with your hands when silverware is available. – Michael Hardy Apr 24 '13 at 14:29
@MichaelHardy, sorry. – Eleven-Eleven Apr 24 '13 at 14:34
I didn't mean to offend, I was just trying to answer the question. I will keep your comment in mind from here on in and have good formatting! – Eleven-Eleven Apr 24 '13 at 14:58
To read postings on is to learn that some people have limited knowledge of $\TeX$ and $\LaTeX$. (My own knowledge of those is quite limited, but I try to be aware of standard conventions.) – Michael Hardy Apr 24 '13 at 15:00
I appreciate the heads up. I suppose in math since rigor is preferred the same applies to the language here. Thanks. – Eleven-Eleven Apr 24 '13 at 15:44

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