## Related questions with answers

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?

Solution

VerifiedDEFINITIONS

$\textbf{Multiplication rule: }$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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