Combinatorics: Number of ways to split a sample equally and randomly.

Question

I have a sample of n scans, which I divide equally and randomly into two sub-samples of size m=n/2. How many unique sub-samples am I able to make in this manner from the original sample size of n scans, please?

Thanks for any help.

Edit

So, if I have 4 scans a,b,c and d, then the unique, equal combinations of this are:

1. a,b and c,d
2. a,c and b,d
3. a,d and b,c

which is just 4 choose 2 divided by 2 = 3? And therefore on a generic scale n choose n / 2 divided by 2?

Answer 1

As you say, you'll have exactly $$ \frac{1}{2}{n \choose \frac{n}{2}} $$

possible choices. Note that, having selected a specific sample (that is, $\frac{n}{2}$ elements of the possible $n$), this determines the other sub-sample, since we have exactly $\frac{n}{2}$ scans left. There are ${n \choose \frac{n}{2}}$ possible choices of a first sample. However, one must take into account that choosing some sample or it's complement gives the same outcome. A concrete example as in your original question: if I pick $\{c,d\}$ first, this determines the other sub sample as $\{a,b\}$, and the same is achieved when selecting $\{a,b\}$ first. This is why we only count half of the previously proposed cases.