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\begin{center}
  {\large\bf Test 1 for Math 405, Introduction to Combinatorics.
    \vspace{12mm}\\ } 
    {\LARGE\bf Name:}\vrule height .4pt depth 0 pt width 8cm
{\footnotesize \hfill September 22, 2000}\\
\end{center}
Instructions: The test will be 50 minutes in length. 


\begin{enumerate}
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\problem  How many sets of three numbers each can be formed from
the numbers $\{1,2,3,\ldots,19,20\}$ if no 2 consecutive numbers
are to be in the set.
\vfill

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\problem  \hide  Final fall93, Test 1 Fall 95  \endhide  
 
Calculate the (multinomial) coefficient of the monomial $a^{3}b^2c$ in the expansion
of $(a+b+c)^{6}$:   

\vfill\newpage 



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\problem  

How many integers between 1 and 1001 are divisible
by at least one of the numbers  7, 11, 13?

\vfill

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\problem Find a cover by dominoes for the following
board. Alternatively prove that no cover can exist.

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  \put(1,0){\line(1,0){5}}  
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\vfill\newpage 

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\problem  
Give a detailed proof why the formula
$$\sum_{k=0}^n {n\choose k}^2 ={2n\choose n}$$ 
holds for all positive integers $n$.

\newpage 

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\problem  There are 100 people in a queue for a London cinema
(and, being in England, the order of people in the queue never changes).
The cinema lets the people into the show in 5 batches, each batch consisting
of 15 or more persons. In how many ways can the 5 batches be
chosen.

\newpage 
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\problem  Find the number of integral solutions of the equation 
$$
x+y+z=30
$$
subject to the conditions:
$$
5\leq x \leq 15,\hspace{12mm} -3\leq y \leq 12, \hspace{12mm} 0\leq z\leq 20.
$$



\newpage 
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\problem At a concert 100 people check their umbrella. Assume
that after the concert the 100 umbrellas are returned completely
at random to the 100 persons who checked the umbrellas. Find the
probability that at least one person goes happily home with his
own umbrella. In your answer provide both a formula and an
estimate in terms of a numerical value. (Like: Formula is
$\frac{\pi}{10}$ or about 0.314).


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\end{enumerate}
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