Rule of product

In combinatorics, the rule of product or multiplication principle is a basic counting principle (a.k.a. the fundamental principle of counting). Stated simply, it is the idea that if we have a ways of doing something and b ways of doing another thing, then there are a · b ways of performing both actions.^[1]

$\begin{matrix} & \underbrace{ \left\{A,B,C\right\} } & & \underbrace{ \left\{ X,Y\right\} } \\ \mathrm{To}\ \mathrm{choose}\ \mathrm{one}\ \mathrm{of} & \mathrm{these} & \mathrm{AND}\ \mathrm{one}\ \mathrm{of} & \mathrm{these} \end{matrix}$

$\begin{matrix} \mathrm{is}\ \mathrm{to}\ \mathrm{choose}\ \mathrm{one}\ \mathrm{of} & \mathrm{these}. \\ & \overbrace{ \left\{ AX, AY, BX, BY, CX, CY \right\} } \end{matrix}$

In this example, the rule says: multiply 3 by 2, getting 6.

The sets {A, B, C} and {X, Y} in this example are disjoint, but that is not necessary. The number of ways to choose a member of {A, B, C}, and then to do so again, in effect choosing an ordered pair each of whose components is in {A, B, C}, is 3 × 3 = 9.

In set theory, this multiplication principle is often taken to be the definition of the product of cardinal numbers. We have

$|S_{1}|\cdot|S_{2}|\cdots|S_{n}| = |S_{1} \times S_{2} \times \cdots \times S_{n}|$

where $\times$ is the Cartesian product operator. These sets need not be finite, nor is it necessary to have only finitely many factors in the product; see cardinal number.

Simple example

When you decide to order pizza, you must first choose the type of crust: thin or deep dish (2 choices). Next, you choose the topping: cheese, pepperoni, or sausage (3 choices).

Using the rule of product, you know that there are 2 × 3 = 6 possible combinations of ordering a pizza.

See also

• Combinatorial principle
• Rule of sum

References

1. ^ http://www.wtamu.edu/academic/anns/mps/math/mathlab/col_algebra/col_alg_tut55_count.htm

This combinatorics-related article is a stub. You can help Wikipedia by expanding it.v · Categories: Combinatorics Combinatorics stubs Wikimedia Foundation. 2010. Игры ⚽ Нужно сделать НИР? Death marches (Holocaust) Fischer Air Look at other dictionaries: Leibniz rule (generalized product rule) — In calculus, the Leibniz rule, named after Gottfried Leibniz, generalizes the product rule. It states that if f and g are n times differentiable functions, then the n th derivative of the product fg is given by:(f cdot g)^{(n)}=sum {k=0}^n {n… …   Wikipedia Product rule — For Euler s chain rule relating partial derivatives of three independent variables, see Triple product rule. For the counting principle in combinatorics, see Rule of product. Topics in Calculus Fundamental theorem Limits of functions Continuity… …   Wikipedia product — /prod euhkt, ukt/, n. 1. a thing produced by labor: products of farm and factory; the product of his thought. 2. a person or thing produced by or resulting from a process, as a natural, social, or historical one; result: He is a product of his… …   Universalium Product integral — Product integrals are a counterpart of standard integrals of infinitesimal calculus. They were first developed by the mathematician Vito Volterra in 1887 to solve systems of linear differential equations. Since then, product integrals have found… …   Wikipedia Product liability — is the area of law in which manufacturers, distributors, suppliers, retailers, and others who make products available to the public are held responsible for the injuries those products cause. Product liability in the United StatesIn the United… …   Wikipedia product liability — liability in tort or delict in respect of produce. Essentially an application of the law of tort although the term can be used to cover liability under sale and supply of goods. Although there is now a Europe wide regime of strict liability, the… …   Law dictionary product rule — Rule for finding the derivative of a product of two functions. If both f and g are differentiable, then (fg)′ = fg′ + f′g. * * * …   Universalium rule of three — n. Math. the method of finding the fourth term of a proportion when three terms are given: the product of the first and last terms is equal to the product of the second and third …   English World dictionary Rule-developing experimentation — See also: Conjoint analysis, Conjoint analysis (in healthcare), Design of experiments. Rule developing experimentation (RDE) is a systematized solution oriented business process of experimentation that designs, tests, and modifies alternative… …   Wikipedia rule of three — a rule in mathematics: the product of the means in a proportion equals the product of the extremes used for finding the fourth term of a proportion where three are given * * * Math. the method of finding the fourth term in a proportion when three …   Useful english dictionary 18+ © Academic, 2000-2024 Contact us: Technical Support, Advertising Dictionaries export, created on PHP, Joomla, Drupal, WordPress, MODx. Mark and share Search through all dictionaries Translate… Search Internet Share the article and excerpts Direct link … Do a right-click on the link above and select “Copy Link”