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\centerline{\bf Midterm Two, April 20 8:00PM-10:15PM on Zoom}
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\centerline{\bf WARNING: THIS MID IS THREE PAGES LONG!!!!!!!!!!!!!!!!!}
\begin{enumerate}
\item
(15 points)
\begin{enumerate}
\item
(5 points)
What is the coefficient of $x^2y^3z^4$ in
$$(x+y+z)^9$$
\item
(10 points)
What is the coefficient of $x^ay^bz^c$ in
$$(x+y+z)^{a+b+c}$$
\end{enumerate}
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\item
(20 points--5 points each)
The Narns play card games with a cards that have ranks in the set $\{1,2,\ldots,r\}$
and suites in the set $\{1,\ldots,s\}$.
In Narn Poker, each player gets $h$ cards.
We assume that both $r$ and $s$ are squares, so $\sqrt{r}$ and $\sqrt{s}$ are
natural numbers.
\begin{enumerate}
\item
A {\it Square Hand} is a hand where all of the cards have square rank.
What is the probability of getting a Square Hand?
(Its okay if they are of the same suite, or not.)
\item
A {\it Square Flush} is a hand where all of the cards have a square rank and all of the suites are the same.
What is the probability of getting a Square Flush?
\item
An {\it Apple} is when you get two of the same rank. There are no other restrictions,
so for example, if you had 3 of the same rank, that would still be an Apple.
What is the probability of getting an Apple.
\end{enumerate}
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\item
(15 points)
In this problem we guide you through the
birthday paradox with $m$ balls in $n$ boxes
where we want the probability that at least $k$ balls
go in the same box is $\ge \frac{1}{2}$.
(HINT: Follow the proof for THREE balls in a box
and feel free to use the approximations I use there.)
Assume that $m$ is much less than $n$.
Assume that $k$ is much less than both $n,m$.
We put $m$ balls into $n$ boxes at random.
\begin{enumerate}
\item
Let $i_1,\ldots,i_k$ be $k$ balls. What is the probability they
are all in the same box?
\item
What is the (approx) probability that NO set of $k$ is in the same box?
(Use three approximations here: (a) that the events are independent, and
(b) use $(1-x)$ is approximately $e^{-x}$, and (c) $\binom{m}{k}\sim \frac{m^k}{k!}$.
\item
Think of $n,k$ as being fixed but $m$ as being varrying.
Approximatly how large does $m$ have to be so that the prob
that $k$ are in the same box is $\ge \frac{1}{2}$?
\end{enumerate}
\end{enumerate}
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