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Name (Last, First): $\underline{\hspace{3in}}$ \hfill Student ID: $\underline{\hspace{2in}}$

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{\small
  \href{\myurl}{Dror Bar-Natan}:
  \href{\myurl/classes/}{Classes}:
  \href{\myurl/classes/\#1415}{2014-15}:
  \href{\myurl/classes/15-475-ProblemSolving}{MAT 475 Problem Solving Seminar}:
  \hfill\url{http://drorbn.net/15-475}
}

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{\large\bf Quiz 3} ``Formulate an Equivalent Problem'', January 27, 2015. You have 25 minutes to solve as much as you can of the following problems. Please write on both sides of the page. \hfill {\bf Good Luck!}

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\noindent{\bf Problem 1.} In how many ways can a natural number $n$ be written as a sum of $k$ non-negative integers, taking order into account? For example, if $n=2$ and $k=3$, there are $6$ ways: $2=2+0+0=0+2+0=0+0+2=0+1+1=1+0+1=1+1+0$.

\noindent{\bf Problem 2} (Larson's 1.3.15). Use a counting argument to prove that for integers $0<r\leq n$,
\[ \binom{r}{r} + \binom{r+1}{r} + \binom{r+2}{r} + \dots + \binom{n}{r} = \binom{n+1}{r+1}. \]

\noindent{\bf Problem 3} (no credit, yet the best solutions will be advertised). What is your favourite ``Formulate an Equivalent Problem'' problem?

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Problem 3 solution by Alessandra:

\[ \includegraphics[scale=0.7]{Quiz3Problem3ByAlessandra.jpg} \]

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Problem 3 solution by Yizhou:

\[\includegraphics[scale=0.7]{Quiz3Problem3ByYizhou.jpg} \]

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