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# A student council consists of three freshmen, four sophomores, four juniors, and five seniors. How many committees of eight members of the council contain at least one member from each class?

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DEFINITIONS

$\textbf{Multiplication rule: }$If an operation consists of $k$ steps, where the first step can be performed in $n_1$ ways, the second step can be performed in $n_2$ ways, ... and the $k$th step can be performed in $n_k$ ways, then the operation can be performed in $n_1n_2...n_k$ ways.

Definition $\textbf{permutation}$ (order is important):

$P(n,r)=\dfrac{n!}{(n-r)!}$

Definition $\textbf{combination}$ (order is not important):

$C(n,r)=\left(\begin{matrix}n\\ r\end{matrix}\right)=\dfrac{n!}{r!(n-r)!}$

with $n!=n\cdot (n-1)\cdot ...\cdot 2\cdot 1$.

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