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Let's say I have a set $S$ and I want all subsets that have two elements. Is there a special name for that?

To put it another way, I want to know if there is a name of the subset of a $S$'s power set that have $n$ elements.

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$k\phantom{}$-subsets? –  J. M. Nov 2 '11 at 16:41
    
The number is determined by a binomial coefficient, C(n,r) giving the number of subsets of order r of a set with n elements. Outside of saying the number of doubletons for your example with r=2, I don't know of any special name attached. Of course if the cardinality of S is infinite, then everything I've said above is not very useful. –  Chris Leary Nov 2 '11 at 16:41
    
@Joe - J.M. has it, I think. It's certainly a good name, even if by some chance it's not standard. –  Chris Leary Nov 2 '11 at 16:43
    
I've seen it notated $\binom{S}{2}$, but I don't think that's a standard notation. –  Henning Makholm Nov 2 '11 at 16:44

2 Answers 2

up vote 3 down vote accepted

The standard name is something like the (set of) $n$-subsets. Less common is $n$-combinations.

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The number of subsets of size $k$ in a set of size $n$ is often denoted $\dbinom nk$, called "$n$ choose $k$".

The set of all subsets of size $k$ in a set $S$ of size $n$ is sometimes denoted $\dbinom Sk$. For example, $\dbinom{\{a,b,c,d\}}{2} = \{\ \{a,b\}, \{a,c\}, \{a,d\}, \{b,c\}, \{b,d\}, \{c,d\} \ \}$.

I sometimes just call them "size-$k$ subsets of $S$".

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