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Homerule of product

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# 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.

## Mathematics Subject Classification

05A05*no label found*03-00

*no label found*

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