\input /home/pasky/school/fastex/lib.tex

\subchapter{Multinomick� v�ta}{

$$ \forall{n} \in{} \nat_{0}, \forall{m} \in{} \nat{}, \forall{x_{1},\ldots,x_{m}} \in{} \real{}: $$

$$ (x_{1}+x_{2}+\cdots+x_{m})^n = \sum_{\scriptstyle k_{1},k_{2},\ldots,k_{m}\ge{}0\atop \scriptstyle k_{1}+k_{2}+\cdots+k_{m}=n} \cn{n}{k_{1},k_{2},\ldots{},k_{m}} (x_{1}^{k_{1}}x_{2}^{k_{2}}\cdots{}x_{m}^{k_{m}}) $$

Bez d�kazu.

\note{
Pro $m=2$ odpov�d� binomick� v�t�.
}

}


\subchapter{Multinomick� koeficient}{

$$ \cn{n}{k_{1},\ldots,k_{m}} = {n!\over{}k_{1}!k_{2}!\cdots{}k_{n}!} $$

Po�et zp�sob� za�azen� ��sel $1,\ldots,n$ do $m$ mno�in $x_{1},\ldots,x_{m}$ tak, aby
$|x_{1}| = k_{1},\ldots,|x_{m}| = k_{m}$.


\scpart{Plat�}{
 $$\cn{n}{k} = \cn{n}{k,n-k}$$
}


\example{
 200 d�t�, 3 autobusy $(80,70,50)$, po�et mo�nost� rozm�st�n�

$$\cn{200}{80, 70, 50}$$

}

}

\bend
