Question

Use Algorithm 3 to list all the 3 -combinations of\({\rm{\{ 1,2,3,4,5\} }}\).

Short answer

All the 3 - combination of\({\rm{\{ 1,2,3,4,5\} }}\)are:

\({\rm{\{ 1,2,3\} ,\{ 1,2,4\} ,\{ 1,2,5\} ,\{ 1,3,4\} ,\{ 1,3,5\} ,\{ 1,4,5\} ,\{ 2,3,4\} ,\{ 2,3,5\} ,\{ 2,4,5\} ,\{ 3,4,5\} }}\)

Step-by-step solution

Definition of Concept

Permutations: A permutation of a set is a loosely defined arrangement of its members into a sequence or linear order, or, if the set is already ordered, a rearrangement of its elements, in mathematics. The act of changing the linear order of an ordered set is also referred to as "permutation."

Lexicographic order: The lexicographic or lexicographical order (also known as lexical order or dictionary order) in mathematics is a generalisation of the alphabetical order of dictionaries to sequences of ordered symbols or, more broadly, elements of a totally ordered set.

List all the 3 –combinations of the given set using Algorithm 3

Considering the given information:

The set\({\rm{\{ 1,2,3,4,5\} }}\).

Using the following concept:

r-Combinations\(\left\{ {{{\rm{a}}_{\rm{1}}}{\rm{,}}{{\rm{a}}_{\rm{2}}}{\rm{, \ldots \ldots }}{{\rm{a}}_{\rm{r}}}} \right\}\)proper subset of\({\rm{\{ 1,2, \ldots \ldots \ldots n\} }}\)not equal to\({\rm{\{ n - r + 1, \ldots \ldots ,n\} }}\)with\({{\rm{a}}_{\rm{1}}}{\rm{ < }}{{\rm{a}}_{\rm{2}}}{\rm{ < \ldots \ldots \ldots }}..{\rm{ < }}{{\rm{a}}_{\rm{r}}}\).

The total number of integers\({\rm{ = 5}}\).

The combination should consist of three integers.

So, the possible number of ways is\({\rm{5}}{{\rm{C}}_{\rm{3}}}{\rm{ = 10}}\).

Therefore, the required all the 3 - combination of\({\rm{\{ 1,2,3,4,5\} }}\)are:

\({\rm{\{ 1,2,3\} ,\{ 1,2,4\} ,\{ 1,2,5\} ,\{ 1,3,4\} ,\{ 1,3,5\} ,\{ 1,4,5\} ,\{ 2,3,4\} ,\{ 2,3,5\} ,\{ 2,4,5\} ,\{ 3,4,5\} }}\)