# theorem about number of sequences - proof and name

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.

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sometimes called the fundamental theorem of counting –  yoyo Aug 12 '11 at 20:43
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 $n$. –  Brian M. Scott Aug 12 '11 at 20:51
I have the most important part - name of the theorem.Thank You. –  exTyn Aug 12 '11 at 21:10
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. –  Arturo Magidin Aug 12 '11 at 21:42
Unfortunately, "product rule" in another context means $(fg)'=fg'+f'g$. –  Gerry Myerson Aug 12 '11 at 22:19