Combinator
 This tool computes combinations, permutations, circulations, and inversions.
 Just enter a set of n items, taken r items at a time, where n >= r; i.e., d = n  r >= 0.

Combinations
A combination is an arrangement from a set of n items taken r items at a time, where ordering does not matter. The number of possible combinations is
c(n, r) = n!/(r!(n  r)!) = n!/(r!d!)
If d = n  r = 0, then c(n, r) = n!/r! = n!/n! = 1.  Permutations
A permutation is an arrangement from a set of n items taken r items at a time, where ordering does matter. The number of possible permutations is
p(n, r) = n!/(n  r)! = n!/d!
If d = n  r = 0, then p(n, r) = n!.  Circulations
A circulation or circular permutation is a circular arrangement from a set of n items taken r items at a time, where relative ordering does matter. The number of possible circulations is
cp(n, r) = n!/(r(n  r)!) = n!/(r(d!))
If d = n  r = 0, then cp(n, r) = n!/r = n!/n.  Inversions
An inversion is a permutation in which the relative order of two items is contrary to that of a reference permutation. The maximum number of inversions in a permutation is
i_{maximum} = n(n  1)/2
and computed for d = n  r = 0.  The total number of inversions in a set of permutations is
i_{total} = (n(n  1)/4)n!
and computed for d = n  r = 0.
 The maximum number of inversions in a permutation is
 Data miners, researchers, teachers, and students.
 Fiveplace numbers are to be formed by using the digits 1 to 8, inclusive, with the digits sorted in descending order. How many of such numbers can be formed?
 How many permutations can be formed by the first 7 letters of the alphabet taken 3 at a time?
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 If abcdefg is a reference sequence, how many inversions are in the sequence cadbfge and why?
 JUNE consists of four letters. These are painted on four marbles, one on each marble. How many pairs of letters can be formed?
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 A batch of 50 articles contains 5 defective articles. A quality control (QC) tranfer scientist inspects this batch by taking 3 articles at random. What is the probability that the selected articles are not defective?
 A batch of 100 manufactured chemical samples is checked by a QC chemistry inspector, who examines 10 chemical samples selected at random. If none of the 10 chemical samples is defective, he accepts the whole batch. Otherwise, the batch is subjected to further inspection. What is the probability that a batch containing 10 defective chemical samples will be accepted?
 How many edges are in polyhedral crystals consisting of n = 4, n = 6 and n = 8 vertices?
 How many microstates are possible for the valence electrons in the outermost orbitals of carbon?
 Three cards are drawn at random from a full deck. What is the probability of getting a three, a seven and an ace?
 What is the probability of being able to form a triangle from three segments chosen at random from five line segments of lengths 1, 3, 5, 7 and 9? Hint. A triangle cannot be formed if one segment is longer than the sum of the other two.
 How many anagrams can be formed from your first name?
 Handbook of Applied Mathematics for Engineers and Scientists Max Kurtz; McGraw Hill, 1991.
 Random Processes in Physical Systems Charles A. Whitney; Wiley, 1990.
 Probability Theory: A Concise Course Y. A. Rozanov; Dover, 1969.
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