The binomial coefficient [tex] \binom{n}{k} [/tex] express exactly the number of ways you can choose k elements from a set of n elements. It is defined as
[tex] \binom{n}{k} = \dfrac{n!}{k!(n-k)!} [/tex]
where n! express the factorial of n, which is defined as the multiplication of all number from n to 2:
[tex] n! = n\times (n-1) \times (n-2) \times \ldots \times 3 \times 2 [/tex]
So, in your case, the number of subset of cardinality 4 that you can extract from a set of cardinality 6 is
[tex] \binom{6}{4} = \dfrac{6!}{2!4!} = \dfrac{6\times5\times4\times3\times2}{2\times4\times3\times2} = \dfrac{6\times5}{2} = 3\times5=15[/tex]
