# 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^{} |