# Permutations/combinations, number of elements and ways

I'm studying the chapter of counting for my discrete math exam and I'm getting a bit confused with the terms permutations/combinations, elements and ways.

As far as I know:

• Permutations/combinations are an arrangement of ordered/unordered distinct elements.
• Elements are what permutations/combinations are made of.
• When we refer to the number of permutations/combinations of an arrangement, we mean how many ways we can do a permutation/combination of that arrangement.

But then there are also the terms r-permutation and r-combination where $r$ are the elements of an arrangement. So, when we do a permutation $P(n,r)$, what is $n$ then? I thought $n$ was the number of elements of our set.

Here's one of the references I used:

I'm a little bit confused... Could someone clarify this for me?

• I would hope that the book provided definitions and examples for these terms. Can you edit your question to post a particular paragraph or statement that puzzles you? – Ethan Bolker Dec 9 '17 at 18:56
• $R$ should be the number of elements that are permuted and $N$ should be the total number of elements. For example, $P(5, 3)$ means $\frac{5\cdot 4\cdot 3\cdot 2\cdot 1}{2\cdot 1}$ which is basically the number of ways to pick 3 things out of 5 where order does matter. – NL628 Dec 9 '17 at 19:04

Here, as an example, $$P(5, 3) = \frac{5!}{2!}$$ means that out of 5 possible objects, how many ways there are to choose 3 objects where order matters. On the other hand, $$C(5, 3) = \frac{5!}{3!\cdot 2!}$$ means the same thing except order doesn't matter.
In this case, note that the textbook says "where order matters." Hence, $P(N, R)$ is number of ways to choose R objects where order matters and $C(N, R)$ is the number of ways to choose an object where order doesn't matter.