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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 $$~10\cdot C(10,3) = \frac{10!}{7! \cdot 3!} = \frac{10 \cdot 9 \cdot 8}{3 \cdot 2 \cdot 1} = 120.~$$