Edit: The mistake was in me counting (enumerating by hand) which @JMoravitz kindly taught me to do right in the chat. Apologies!
I'm trying to code a problem I have, and I would like to use a combination approach. But I seem to make the wrong assumptions, or I cannot calculate right (which is reasonably plausible).
If I have, n elements and I pick $\frac n2$ elements, I thought the number of combinations I get should be n choose $\frac n2$ ?
For example 4 elements gives 6 combinations. And 6 elements give 20 combinations.
So if I have elements a b c d I can have:
a b x x or
a x c x or
a x x d or
x b c x or
x b x d
total of 6.
If I do this for 6 elements a b c d e f
a b c x x x
a b x d x x
x b c d x x
x b c x e x
x x x d e f
And count them I am getting a total of 16 combinations? Am I missing some with this method? If not (though possible, I went over it numerous times already), how do I calculate the right number of combinations?
Thank you for your help.