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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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    Unfortunately, "product rule" in another context means $(fg)'=fg'+f'g$.2011-08-12

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The following names were given in the comments:

  1. Fundamental theorem of counting

  2. Chinese menu principle

  3. Multiplication rule

  4. Product rule