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\begin{center}
  {\large\bf Final for Math 405, Introduction to Combinatorics.
    \vspace{12mm}\\ } {\LARGE\bf Name:}\vrule height .4pt depth 0
  pt width 8cm
  {\footnotesize \hfill December 12, 2000}\\
\end{center}
Instructions: The test will be 120 minutes in length.


\begin{enumerate}
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  \problem (10 pts) Calculate the (multinomial) coefficient of
  the monomial $a^{4}b^3c$ in the expansion of $(a+b+c)^{8}$:

\vfill

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\problem (10 pts) Give a detailed proof why the formula
$$
\sum_{k=0}^n (-1)^k{n\choose k} =0.
$$
holds for all
positive integers $n$.

\vfill\newpage 
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\problem a) (7 pts)
Find the number of nonnegative integral solutions of the equation 
$$
a+b+c+d=100
$$
\vfill

 a) (8 pts) Same as a) with the additional requirement that
 $a\geq 10$, $a$ is even and $0\leq b\leq 1$.


\vfill\newpage 

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\problem (10 pts) Recall that the Euler $\phi$ function $\phi(n)$
is defined as the number of integers $1\leq m<n$ such that
$\mathrm{gcd}(m,n)=1$. Compute (Hint use inclusion/exclusion
principle):\bigskip

a) (5 pts)\hspace{2cm} $\phi(101)=$ \vfill

b) (5 pts)\hspace{2cm}  $\phi(143)=$ \vfill  

c) (5 pts)\hspace{2cm}  $\phi(1001)=$ 

\vfill\newpage 
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  \problem 
  (10 pts) Assume that
  $$
  h_{n}=3h_{n-1} -2 h_{n-2}
  $$
  for $n\geq 3$ and that $h_1=h_2=1$. Compute $h_n$ for all
  integers $n\geq 3$.


\vfill

%%%%%%%%%%%%%%%%%%%%%%%
  \problem 
  (10 pts) Solve explicitly the recurrence relation
  \begin{eqnarray*}
    h_{n+1}&=&  2h_n+7,\ \ n\geq 1.\\
     h_1&=&2.
  \end{eqnarray*}



\vfill\newpage 
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  \problem \hide Final fall95 \endhide %
  (15 pts) Let
  $$
  f(x)=\frac{1}{x(x-1)(x-2)(x-3)}=\sum_{n=-1}^\infty s_nx^n
  $$
  be the generating function of a sequence. Find an explicit
  formula for $s_n$.  (Hint: Think what the effect of the factor
  $\frac{1}{x}$ might be.)

\vfill\newpage 
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\problem (15 pts) Determine the number of $n$ digit numbers with
all digits odd, such that 1 and 3 each occur an even number of
times.


\newpage
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\problem (10 pts) Consider  the binary $[7,4]$ code having
parity check matrix
$$
H:=  
\left(\begin{array}{ccccccc}
1 & 0 & 0 & 1 & 0 & 1 & 1 \\
0 & 1 & 0 & 1 & 1 & 0 & 1 \\
0 & 0 & 1 & 0 & 1 & 1 & 1 
\end{array}\right) .
$$
Assume one error did happen and the vector $y=(0,1,1,0,0,0,0)$ was
received. Find the code word which was sent.

\vfill

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\problem (10 pts) Let $\F:=\{ 0, 1,2\}$ be the field of three
elements. Count the number of 2-dimensional subspaces in $\F^4$.


\vfill\newpage 
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\problem (15 pts) Let $\F:=\F_{11}=GF(11)$ be the Galois field of
11 elements.  Construct an ISBN-code $C\subset \F^{10}$ which
contains more than 200'000'000 code-words (ISBN-numbers) and
which is capable of correcting one error. (Note that the ISBN
system in use only can detect an error and there is no way of
correcting an error).



\vfill\newpage 
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\problem  (15 pts) Alice publishes the encryption function 
$$
\varphi: \Z_{33}\rightarrow \Z_{33},\ \ x\longmapsto x^7.
$$
Bob, not aware that everybody can compute the factorization of
$n=33$ sent to Alice the message $17=x^7$. Find the message $x$
of Bob. Note: You should not use trial and error and try out all
32 possibilities.



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\end{enumerate}
\end{document}


