Combinatorics

combinatorics

[‚kəm·bə·nə′tȯr·iks]

(mathematics)

Combinatorial topology which studies geometric forms by breaking them into simple geometric figures.

McGraw-Hill Dictionary of Scientific & Technical Terms, 6E, Copyright © 2003 by The McGraw-Hill Companies, Inc.

The following article is from The Great Soviet Encyclopedia (1979). It might be outdated or ideologically biased.

Combinatorics

 

(1) The same as mathematical combinatorial analysis.

(2) A branch of elementary mathematics concerned with the study of the number of combinations, obeying some condition or other, that may be formed out of a given finite set of objects (of any nature, for example, letters, numbers, arbitrary symbols).

The following are the most useful formulas of combinatorics.

(1) Number of arrangements. In how many ways can one select m objects from n different objects (taking into account the order in which the objects are selected)? The number of ways is equal to

A_n^m is said to be the number of arrangements of n elements taken m at a time.

(2) Number of permutations. Let us examine the following problem: in how many ways can one order n different objects? The number of ways is equal to

P_n = 1 · 2 · 3 . . . n = n!

(The symbol n! is read “n factorial”; it is also convenient to regard 0! as being equal to 1.) P_n is said to be the number of permutations of n elements.

(3) Number of combinations. In how many ways can one select m objects from n different objects (ignoring the order in which the objects are selected)? The number of ways is equal to

C_n^m is said to be the number of combinations of n elements taken m at a time. The numbers C_n^m can be obtained as the coefficients in the expansion of the n th power of a binomial:

Therefore they are also called binomial coefficients. Some basic relations of binomial coefficients are

The connection between the numbers A_n^m, P_m, and C_n^m is given by the relation

Arrangements with repetition (that is, all possible choices of m objects of n different types, the order of choice being essential) and combinations with repetition (analogue of previous concept but without consideration of order) are also considered. The number of arrangements with repetition is n^m, while the number of combinations with repetition is

Some fundamental rules for solving combinatorial problems follow.

(1) The sum rule. If a certain object A can be selected in m ways from a set of objects and if another object B can be selected in n ways, then there are m + n possibilities of choosing either A or B.

(2) The product rule. If an object A can be chosen in m ways and if after each such choice the object B can be chosen in n ways, then the ordered pair (A, B) can be chosen in mn ways.

(3) The principle of inclusion and exclusion. Let N objects be given possessing n properties α₁, α₂, . . . , α_n. Let N (α_i, α_j, . . . , α_k) denote the number of objects possessing the properties α_i, α_j, . . . ,α_k as well as perhaps other properties. Then the number N’ of objects that do not possess any of the properties α₁, α₂, . . . , α_n is given by the formula

N’ = N – N(∞₁) – N(∞₂) – . . . – N(∞_n)

+ N(∞₁, ∞₂) + N(∞₁, ∞₃ + . . . + N(∞_{n – 1}, ∞_n)

– N(∞₁, ∞₂, ∞₃) – N(∞_{n – 2}, ∞_{n – 1}, ∞_n

+ . . . + (–1)ⁿN(∞₁, . . . , ∞_n

REFERENCE

Netto, E. Lehrbuch der Combinatorik, 2nd ed. Leipzig-Berlin, 1927.

V. E. TARAKANOV

The Great Soviet Encyclopedia, 3rd Edition (1970-1979). © 2010 The Gale Group, Inc. All rights reserved.