# 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,

 $\mbox{card}(A\!\times\!B)\;=\;m\!\cdot\!n.$

The rule of product is true also for the combination of several processes:  If the processes $A_{i}$ can have $n_{i}$ possible results ($i=1,\,2,\,\ldots,\,k$), then their combined process has $n_{1}n_{2}\!\cdots\!n_{k}$ possible results.  I.e.,

 $\mbox{card}(A_{1}\!\times\!A_{2}\!\times\ldots\times\!A_{k})\;=\;n_{1}n_{2}\!% \cdots\!n_{k}.$

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

 $n(n\!-\!1)(n\!-\!2)\!\cdots\!2\!\cdot\!1\;=\;n!$

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