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\documentclass[11pt]{exam}
\input{defs}
\begin{document}
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\cl {\ub {\bf Pascal's Triangle } } 
\noi

\cl{
\includegraphics[height=2.5in,width=3.5in]{/home/dwick/DATA/Professional/Other/Pictures&Worksheets/Pascals_Triangle_1-10.pdf}
}
\begin{questions}
\question
If we consider the top row as row $0$, then the $k+1$ entries in the $k^{th}$ row are
$${k\choose0} \  \ {k\choose1} \ldots \ldots{k\choose k-1} \ \ {k\choose k}$$
\question
Each entry is the sum of the two entries directly above it: For example, $56=21+35$. This yields the generating formula
$${n\choose k}={n-1\choose k-1}+{n-1\choose k}$$
\question
The sum of all entries in the $k^{th}$ row is $2^k$, or
$${k\choose0} + {k\choose1} + \ldots +{k\choose k-1} +{k\choose k}=2^k$$
\question
The entries in the third diagonal are the triangular numbers $1,3,6,10,\ldots$
\question
Every other entry in the third diagonal gives the hexagonal numbers $1,6,15,28,\ldots$
\question
The entries in the fourth diagonal are the tetrahedral numbers $1,4,10,20,35,\ldots$
\question If the second entry in a row is a prime number $p$, then every entry in that row (greater than one) is divisible by $p$.
\question
If we consider the entries in a row as a base 10 number (allowing for digits greater than 9), then the $k^{th}$ row represents the number $11^k$. 
For example, for row $k=4$ we have
$$1\times 10^4+4\times 10^3+6\times 10^2+4\times 10^1+1\times 10^0=14641=11^4$$
and for row $k=6$ we have
$$1\times 10^6+6\times 10^5+15\times 10^4+20\times 10^3+15\times 10^2+6\times 10^1+1\times 10^0=1771561=11^6$$
This yields the formula
$${k\choose k} 10^k + {k\choose k-1}10^{k-1} + \ldots +{k\choose 1} 10^1 +{k\choose0 } 10^0=11^k$$
\question
If we sum the numbers in any diagonal, then the sum is the number below and to the right of the last entry in the sum. For example, $1+3+6+10=20$. This yields the formula
$${n\choose k}={n-1 \choose k-1}+{n-2\choose k-1}+\ldots +{k\choose k-1}+{k-1\choose k-1}$$
A schematic of this property is shown below

\cl{\includegraphics[height=3in,width=4in]{/home/dwick/DATA/Professional/Other/Pictures&Worksheets/Pascals_Triangle_1-10.pdf}}

\bigskip
\bigskip
\question
The Fibonacci numbers occur by adding the entries according to the schematic below

\cl{\includegraphics[height=3in,width=4in]{/home/dwick/DATA/Professional/Other/Pictures&Worksheets/Pascals_Triangle_1-10.pdf}}
\vspace{-3.0in}
%\hspace{4.5in}
$2=1+1$ \\
$3=2+1$ \\
$5=1+3+1$ \\
$8=3+4+1$ \\
$13=1+6+5+1$ \\
$21=4+10+6+1$ \\
$34=1+10+15+7+1$ \\
{\it etc}

\end{questions}
\end{document}