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Permutations and combinationsthe various ways in which objects from a set may be selected, generally without replacement, to form subsets.
This selection of subsets is called a permutation when the order of selection is a factor, a combination when order is not a factor. By considering the ratio of the of desired subsets to the of all possible subsets for many games of chance in the 17th century, the French mathematicians Blaise Pascal and Pierre de Fermat gave impetus to the development of combinatorics and probability theory.
The concepts of and differences between permutations and combinations can be illustrated by examination of all the different ways in which a pair of objects can be selected from five distinguishable objects—such as the letters A, B, C, D, and E.
If both the letters selected and the order of selection are considered, then the following 20 outcomes are possible:.
Quantitative flavors: continuous data and discrete data
Each of these 20 different possible selections is called a permutation. The expression n!
For example, using this formula, the of permutations of five objects taken two at a time is. Thus, for 5 objects there are 5! For combinations, k objects are selected from a set of n objects to produce subsets without ordering. This is the same as the nk binomial coefficient see binomial theorem ; these combinations are sometimes called k -subsets.
For example, the of combinations of five objects taken two at a time is. The formulas for n P k and n C k are called counting formulas since they can be used to count the of possible permutations or combinations in a given situation without having to list them all.
Permutations and combinations. Additional Info.
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The two main flavors of data: qualitative and quantitative
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Subscribe Now. Learn More in these related Britannica articles:. A permutation of a set, say the elements aband cis any re-ordering of the elements, and it is usually denoted as follows:….
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