# Combinations and Permutations

In order to attain the solution to several statistical experiments, you need to be able to count the number of points in a sample space. However, counting these points could turn out to be hard and tedious. We are but fortunate and lucky enough to have various methods to make this task of counting easier. Combinations and permutations are probably among those.

Combinations and Permutations

When you’ve been given the number of objects to choose from and you’re required to find the number of ways to find the possible number of ways of doing so, you would be obliged to deal with combinations and permutations.

## Combination

A combination is known to be an unordered collection of several items from a set. While combinations are known to be unordered collections, any alteration in the way of arrangements of various elements would not determine a new combination. As per the combination formula

The number of ways to select $k$ objects from the given group of $n$ objects, where order of the elements does not matter is:

$nC_{k}=\binom{n}{k}=\frac{n!}{k!(n-k)!}$

Some points regarding combinations:

• Generally, any considered $n$ objects could be arranged in $n(n-1)(n-2)\ldots .(3)(2)(1)$ ways. This product is represented by $n!$ and is addressed as n factorial.
• A combination ($nC_{k}$) could be expressed as a selection of either all or only a part of the set of objects that is given without considering the order in which they have been selected. Thus any combination XYZ would be dealt as same combination even when written as ZYX.

## Permutation

A permutation, on the other hand, is known to be an ordered arrangement with several or all the elements from a set. Since permutations are ordered arrangements, any alterations in the order of arrangement of the elements would be regarded as a new permutation. The mathematical formula as per permutation for the number of ways to select $k$ objects from the considerable group of $n$ objects is:

$nP_{k}=k!\binom{n}{k}=k! \frac{n!}{k!(n-k)!}=\frac{n!}{(n-k)!}$

Some points regarding permutations:

• A permutation could be expressed as an arrangement either all or a part of the set of given objects, considering the order of arrangement of the elements. This implies that XYZ will be considered as a different permutation when written as YZX.
• The number of permutations for selecting k objects from a set of n objects is represented by $nP_{k}$.

Based on the above concepts, counting decisions could be made easier with combinations and permutations.