![]() ![]() N! we call the factorial of the number n, which is the product of the first n natural numbers. For example, if we have the set n = 5 numbers 1,2,3,4,5, and we have to make third-class variations, their V 3 (5) = 5 * 4 * 3 = 60. The number of variations can be easily calculated using the combinatorial rule of product. The elements are not repeated and depend on the order of the group's elements (therefore arranged). In computer security, if you want to estimate how strong a password is based on the computing power required to brute force it, you calculate the number of permutations, not the number of combinations.C k ( n ) = ( k n ) = k ! ( n − k ) ! n ! n = 1 0 k = 4 C 4 ( 1 0 ) = ( 4 1 0 ) = 4 ! ( 1 0 − 4 ) ! 1 0 ! = 4 ⋅ 3 ⋅ 2 ⋅ 1 1 0 ⋅ 9 ⋅ 8 ⋅ 7 = 2 1 0 The number of combinations: 210Ī bit of theory - the foundation of combinatorics VariationsĪ variation of the k-th class of n elements is an ordered k-element group formed from a set of n elements. For example, a "combination lock" is in fact a "permutation lock" as the order in which you enter or arrange the secret matters. Very often permutations are mistaken for combinations, at least in common language use. ![]() For example, if you are thinking of the number of combinations that open a safe or a briefcase, then these are in fact permutations, since changing the order of the numbers or letters would result in an invalid code. Permutations are for ordered lists, while combinations are for unordered groups. With combinations we do not care about the order of the things resulting in fewer combinations. The difference between combinations and permutations is that permutations have stricter requirements - the order of the elements matters, thus for the same number of things to be selected from a set, the number of possible permutations is always greater than or equal to the number of possible ways to combine them.
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