rule of product

If a process $A$ can have altogether $m$ different results and another process $B$ altogether $n$ different results, then the two processes can have altogether $mn$ different combined results.  Putting it to set-theoretical form,

$$\text{card}(A\times B)=m\cdot n.$$

The rule of product is true also for the combination of several processes:  If the processes $A_i$ can have $n_i$ possible results ($i=1,\mathrm{\hspace{0.17em}2},\mathrm{\dots },k$), then their combined process has $n_1{n}_2\mathrm{\cdots }{n}_k$ possible results.  I.e.,

$$\text{card}(A_1\times A_2\times \mathrm{\dots }\times A_k)=n_1{n}_2\mathrm{\cdots }{n}_k.$$

Example.  Arranging $n$ elements, the first one may be chosen freely from all the $n$ elements, the second from the remaining $n-1$ elements, the third from the remaining $n-2$, and so on, the penultimate one from two elements and the last one from the only remaining element; thus by the rule of product, there are in all

$$n(n-1)(n-2)\mathrm{\cdots }2\cdot 1=n!$$

different arrangements, i.e. permutations, as the result.

Title	rule of product
Canonical name	RuleOfProduct
Date of creation	2013-03-22 19:13:02
Last modified on	2013-03-22 19:13:02
Owner	pahio (2872)
Last modified by	pahio (2872)
Numerical id	6
Author	pahio (2872)
Entry type	Definition
Classification	msc 05A05
Classification	msc 03-00
Synonym	multiplication principle
Related topic	CartesianProduct
Related topic	Combinatorics
Related topic	Cardinality
Related topic	Number
Related topic	Product