JEE Main & Advanced Mathematics Permutations and Combinations Definition

Each of the different groups or selections which can be formed by taking some or all of a number of objects, irrespective of their arrangements, is called a combination.

Notation : The number of all combinations of \[n\] things, taken \[r\] at a time is denoted by \[C\,(n,\,r)\] or \[^{n}{{C}_{r}}\,\text{or }\left( \begin{matrix} n  \\ r  \\ \end{matrix} \right)\]. \[^{n}{{C}_{r}}\] is always a natural number.

Difference between a permutation and combination :

(i) In a combination only selection is made whereas in a permutation not only a selection is made but also an arrangement in a definite order is considered.

(ii) Each combination corresponds to many permutations. For example, the six permutations \[ABC,\text{ }ACB,\text{ }BCA,\text{ }BAC,\text{ }CBA\] and \[CAB\] correspond to the same combination \[ABC\].