Combinations and permutations are widely used in probability theory and math statistics. In this context, combination is defined as un-ordered collection of distinct elements taken from the finite set. In other words, combinations is the number of all possible ways the finite set of N elements could be split into smaller groups of M elements, where order of elements in the groups does not matter. Permutation is the ordered collection of distinct elements taken from the finite set, or, in other words it is the number of all possible ways the finite set of N elements could be split into smaller groups of M elements, where order of elements really matter. You could think of permutations as the ordered combinations.
Practical computation of both combinations and permutations is heavily based on the factorial function. As a reminder, following are the main properties of the factorial function, shown as sample numeric and general symbolic representations:
0! = 1 (this seems weird, but it’s a matter of convention)
1! = 1
K! = 1 * 2 * ... K
2! = 1 * 2 = 2
3! = 1 * 2 *3 = 6
In general, the factorial of any natural number is a product of all smaller numbers and the number itself (0 is obviously excluded from this generalization).
Combinations are calculated using factorial function as follows: assuming that the finite set contains N element total, then the number of possible M-elements combinations is:
C(M,N) = N!/(M!*(N-M)!).
Corresponding numeric example: C(4,3) = 4!/(3!*(4-3)!) = 24/(6*1) = 4
Permutations can be calculated using the formula:
P(M,N) = N!/(N-M)!
Corresponding numeric example: P(4,3) = 4!/(4-3)! = 24
Functional dependency between combinations and permutations is described by simple formula:
P(M,N) = C(M,N) * M!
In general, number of permutation is always more or equal to number of combinations:
P(M,N) => C(M,N) * M!
Combinations and permutations both belong to the mathematical domain referred as combinatorics. In order to calculate combinations and permutations, students could use the free online combinatorics calculator (see the sample screenshot attached to the article). The usage is rather straightforward: enter the total number of elements N and the number of elements M in selection into the corresponding text boxes and click on “solve” screen button to get the result. For example, assuming N=5 and M=3, then the number of all possible combinations C=10 and permutations P=60.
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