Question

Another Probability/General Counting Question:
An airport van has twelve passengers. It stops at seven downtown hotels, and each of the passengers gets off at one of them.
a) How many possible tally sheets can the driver make showing how many passengers get off at each hotel?
b) How many possible tally sheets can he make with at least one hotel receiving no passengers?

(work to follow)

Answer 1

I believe for part a), it is simply
\[\left(\begin{matrix}12+7-1 \\ 12\end{matrix}\right)\]
or \[\left(\begin{matrix}18 \\ 12\end{matrix}\right) = 18,564 \]

Answer 2

For part b), I believe it is
\[\left(\begin{matrix}17 \\ 12\end{matrix}\right)+\left(\begin{matrix}16 \\ 12\end{matrix}\right) + ...+ \left(\begin{matrix}12 \\ 12\end{matrix}\right)\]
= 6,188+1,820+455+91+13+1 = 8,568.

Anyone feel like checking my work?