Combinations and Permutations

The study of combining and rearranging items as well as their arrangement in a set is known as permutation and combination. While combinations can be represented by a specific series of combination numbers, permutations can be represented by a particular permutation number sequence. Without actually naming the items, permutation and combination are two methods of counting that assist us in figuring out how many distinct arrangements and selections there are for a given set of items.

Permutation

The organisation of a group of individuals or items into a sequence or order is known as a permutation. In the case of permutations, order matters. Ordered configurations of items are called permutations. Order is taken into account, therefore permutations with the same elements but different orders are thought to be separate. An r-permutation is a permutation that only comprises a portion of the elements in a particular set.

Permutation Formulas:

The number of a set of n items taken r at a time, is denoted by P_rⁿ

Formula will be :

 P_rⁿ= n! / n-r!

The number of permutations of n unique things taken r at a time, where r is the number of arrangements where a specific object is included. r.P_r-1^n-1

When an object is consistently excluded from an arrangement, the number of arrangements equals P._r^n-1

There are m! (n – m + 1)! permutations of n distinct items taken all at once when m particular objects always combine.

When m specific objects are not combined, there are n! – m! (n – m + 1) permutations of all n different objects chosen at once.

Combination

Combinations are collections of unsorted items. A combination can contain any number of different elements that are drawn from a set of separate elements. Combinations are sometimes known as r-combinations for this reason. Combination is the process of choosing some or all things from a given collection of alternatives without taking the order of the choices into account.

Combination Formula

The formula is when r items can be made to select among n things simultaneously in a combination.

 C_rⁿ = n! / r!n-r!

Relationship between Combinations and Permutations  C_rⁿ= P_r^{n  /} r

It is important to note that a permutation distinguishes between several orderings, but a combination does not. With enough repetition, permutations and combinations are possible. Whether duplication of elements is permitted will depend on the circumstances.

Derangements

A derangement of the original order occurs when a permutation is created using the same collection of objects but none of the objects are placed in the same order as they were in the original arrangement.

Conclusion

The techniques used to determine how many outcomes are conceivable in certain circumstances are permutation and combination. Combinations and permutations are referred to as selections and arrangements, respectively. The sum rules and product rules, which are based on the basic concept of counting, make counting simple to use. A permutation is an arrangement of several items taken one at a time or all at once in a specific order. Grouping is the foundation of a combination. Combinations are a useful tool for calculating the number of distinct groups that can be created from the provided items.