Combinations are subsets of a given size of a given finite set. All questions for this tag have to directly involve combinations; if instead the question is about binomial coefficients, use that tag.

A combination is a way of choosing elements from a set in which order does not matter.

A wide variety of counting problems can be cast in terms of the simple concept of combinations, therefore, this topic serves as a building block in solving a wide range of problems.

The number of combinations is the number of ways in which we can select a group of objects from a set.

The difference between combinations and permutations is ordering. With permutations we care about the order of the elements, whereas with combinations we don’t.

Notation: Suppose we want to choose $~r~$ objects from $~n~$ objects, then the number of combinations of $~k~$ objects chosen from $~n~$ objects is denoted by $~n \choose r~$ or, $~_nC_r~$ or, $~^nC_r~$ or, $~C(n,~r)~$.

$~n \choose r~$$=\frac{1}{r!}~^nP_r=\frac{n!}{r!~(n-r)!}$

Example: Picking a team of $~3~$ people from a group of $$~10\cdot C(10,3) = \frac{10!}{7! \cdot 3!} = \frac{10 \cdot 9 \cdot 8}{3 \cdot 2 \cdot 1} = 120.~$$