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\begin{document}
\title{{\bf Problem Set 1 Solutions} }
\author{Name: TODO}
\date{}
\maketitle
\section*{Collaborators}
TODO: List collaborators, if any.
\newpage
\section*{Problem 1 (Softball) Solution}
Thirteen people (5 children and 8 adults) on a softball team show up for a game.
\begin{enumerate}
\item (3 points) How
many ways are there to choose 4 players to take the field?
\begin{tcolorbox}
TODO: Your solution here.
\end{tcolorbox}
\item (3 points) How many
ways are there to assign 4 players to the positions of catcher, pitcher, 1st baseman and shortstop by selecting players from the 13 people who show up?
\begin{tcolorbox}
TODO: Your solution here.
\end{tcolorbox}
\item (4 points) How
many ways are there to choose 4 players to take the field if at
least one of these players must be an adult?
\begin{tcolorbox}
TODO: Your solution here.
\end{tcolorbox}
\end{enumerate}
\newpage
\section*{Problem 2 (Counting words) Solution}
We want to count the number of strings of length $6$ from
the English alphabet $\{A, B, \cdots, Z\}$ subject to a number of
different constraints. Note that we consider the English alphabet here
to consist of 6 {\em vowels} ($\{A,E,I,O,U,Y\}$) and 20 {\em
consonants}.
\noindent How many strings are there which ...
\begin{enumerate}
\item (4 points) ... are only made of vowels?
\begin{tcolorbox}
TODO: Your solution here.
\end{tcolorbox}
\item (4 points) ... are only made of consonants?
\begin{tcolorbox}
TODO: Your solution here.
\end{tcolorbox}
\item (4 points) ... have {\em exactly} one vowel?
\begin{tcolorbox}
TODO: Your solution here.
\end{tcolorbox}
\item (4 points) ... have {\em exactly} two vowels?
\begin{tcolorbox}
TODO: Your solution here.
\end{tcolorbox}
\item (4 points) ... have at most two vowels, which may only appear in the second and
fourth position?
\begin{tcolorbox}
TODO: Your solution here.
\end{tcolorbox}
\item (4 points) ... have at least one vowel?
\begin{tcolorbox}
TODO: Your solution here.
\end{tcolorbox}
\end{enumerate}
In all cases, explain your reasoning exactly -- do not just give
numbers or unjustified calculations.
\newpage
\section*{Problem 3 (Arrangements) Solution}
How many different ways are there to arrange the letters in the following words?
\begin{enumerate}
\item (6 points) {\bf MISSISSAUGA}
\begin{tcolorbox}
TODO: Your solution here.
\end{tcolorbox}
\item (6 points) {\bf statistics}
\begin{tcolorbox}
TODO: Your solution here.
\end{tcolorbox}
\end{enumerate}
\newpage
\section*{Problem 4 (Five card hands) Solution}
How many ways are there to select 6 cards from a standard deck
of 52 cards if we require that all 4 suits are represented? Order doesn't matter.
\begin{tcolorbox}
TODO: Your solution here.
\end{tcolorbox}
\newpage
\section*{Problem 5 (From here to there) Solution}
In this problems you will consider paths on the integer grid
that start at (0,0) in which every step
increments one coordinate by 1 and leaves the other unchanged.
\begin{enumerate}
\item (4 points) How many such paths are there from (0,0) to (85, 65)?
\begin{tcolorbox}
TODO: Your solution here.
\end{tcolorbox}
\item (4 points) How many such paths are there from (0,0) to (85, 65) that go through (10,35)?
How many such paths if they must go through (15,40) instead?
\begin{tcolorbox}
TODO: Your solution here.
\end{tcolorbox}
\item (6 points) How many such paths are there from (0,0) to (85, 65) that
go through (10,35), but do \emph{not} go through (15,40).
\begin{tcolorbox}
TODO: Your solution here.
\end{tcolorbox}
\item (6 points) How many such paths from (0,0) to (85, 65) are there that go through neither of (10,35) nor (15,40)?
\begin{tcolorbox}
TODO: Your solution here.
\end{tcolorbox}
\end{enumerate}
\newpage
\section*{Problem 6 (Binomial Theorem applications) Solution}
\begin{enumerate}
\item (7 points) What is the coefficient of $x^4y^{12}$ in the expansion of $(x-3y^3)^{8}$?
\begin{tcolorbox}
TODO: Your solution here.
\end{tcolorbox}
\item (8 points) Use the binomial theorem to prove that
\[ \sum_{i=0}^{300} {300 \choose 300 -i}(-6)^{300-i}
= 5^{300} \]
\begin{tcolorbox}
TODO: Your solution here.
\end{tcolorbox}
\end{enumerate}
\newpage
\section*{Problem 7 (A gentle introduction to Python) Solution}
\begin{enumerate}
\item (10 points) Read the \href{https://edstem.org/us/courses/32027/lessons/51246/slides/286967}{Pset1 Coding} lesson on Edstem and follow the directions to complete 5 coding exercises. Then submit all required files to PSet1 [Coding] on Gradescope. The score that appears on Gradescope for this part is final.
\begin{tcolorbox}
Nothing to write for this question.
\end{tcolorbox}
\item (5 points) Read the \href{https://edstem.org/us/courses/32027/lessons/51270/slides/287088}{Edstem lesson} on Python's numpy library, after completing the previous part. You do \textbf{not} need to complete any coding exercises or submit anything to Gradescope for this part. The exercise that is there is entirely OPTIONAL, and intended only for practice if you need it. Afterwards, write down what you felt was the most confusing numpy function and/or class to you and why. If nothing is confusing, explain which function and/or class is the most interesting to you. We will grade based on completion and effort rather than correctness, and it's recommended that your answer be no longer than 5 sentences.
\begin{tcolorbox}
TODO: Your solution here.
\end{tcolorbox}
\end{enumerate}
\end{document}