Combinations without putting back

Combinations are the number of possible subsets out of a set (note: not ordered; if it were ordered, that would be called Variation).

The number of possible combinations of taking 1 item out of a set of N items is N.

The number of possible combinations C of taking n items out of a set of N items (when putting them back only after taking all the n items) is:

C=\binom{N}{n}:=÷{N!}{n!⋅(N-n)!}

that is, "all!" divided by "good!"⋅"bad!".

Properties

\binom{N}{n}=0,n>N
\binom{N}{0}=1
\binom{N}{1}=N
\binom{N}{N}=1 (because we are talking of an set, which is unordered)
n⋅\binom{N}{n}=N⋅\binom{N-1}{n-1}

Combinations with putting back

TODO