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There is a theorem which says that number of sequences $(x_1, \ldots, x_k, \ldots, x_n)$ where $x_k$ can be choosen in $m_k$ ways, $k = 1, 2, \ldots, n$ is equal to $m_1 \cdot m_2 \cdot \ldots \cdot m_n$.

I'm looking for a proof (or proofs) and official name of this theorem.

Thank you for your help.

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    sometimes called the fundamental theorem of counting2011-08-12
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    Also called the Chinese menu principle, the multiplication rule, and the product rule. Since there’s no body that rules on such things, there cannot be an official name; there is merely common usage. It’s pretty obvious for $n=2$, and from that it follows by induction for larger $n$.2011-08-12
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    I have the most important part - name of the theorem.Thank You.2011-08-12
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    I know it as the **Product Rule**: if one event can occur in $n$ different ways and a second event can occur in $m$ different ways, then the total number of ways in which *both* events can occur is $nm$. See [this previous answer](http://math.stackexchange.com/questions/11307/how-do-i-do-combinatorics-counting/11312#11312).2011-08-12
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    Unfortunately, "product rule" in another context means $(fg)'=fg'+f'g$.2011-08-12

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