# What is the professional term for the combination of the selection in n out of the total m elements?

I know the number of combinations is called ${}_nC_r$, but what about all the exact outcomes?

For example: I have $3$ elements $a,b,c$ and for the parameter $2$, I will have outcomes $$ab,\quad ac,\quad ba,\quad bc,\quad ca,\quad cb$$

I want to search different implementations of this, but I don't know what term should I input in google.

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In french, this is Arrangement (see fr.wikipedia.org/wiki/Arrangement_(math%C3%A9matiques)). – Arnaud May 20 '13 at 9:41
That's a nice information. I don't know why that is french only though >< – Abby Chau Yu Hoi May 20 '13 at 10:09
If you look at the Wikipedia article, you could see that there is also a german word, an italian word... – Arnaud May 20 '13 at 11:05

If I have a set $X$ with $n$ elements, and I wanted to refer one of the possible "outcomes" of choosing $k$ elements (without the order mattering), I would just call it "a $k$-element subset of $X$". Thus, the number of $k$-element subsets of $X$ is $\binom{n}{k}$ or ${}_nC_k$, depending on your notational preferences. I'm not aware of any more specialized term.
I would use the term "$k$-tuple of elements of $X$" to mean an ordered $k$-element subset of $X$ (possibly with repetitions). Thus, the number of $k$-tuples of distinct elements of $X$ is ${}_nP_k$, and the number of $k$-tuples of elements of $X$ is $n^k$.