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{\bf MATH304  \hspace{1cm} HW 5 - midterm test\hfill \name}
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Due \textbf{Oct  1} before class, \textbf{answer without justification will receive 0 points}.
The solution has to be typed (using \LaTeX). 
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\question{}{
Prove that in a group of $n > 1$ people there are two who have the same number of 
acquaintances in the group. (It is assumed that no one is acquainted with oneself.)
}


\textbf{Solution:}


\question{}{
Combinatorially prove the following binomial identity
$$ \sum_{k=0}^{r} {n+k\choose k} = { n + r + 1 \choose r } .$$
}


\textbf{Solution:}


\question{}{
There are 20 identical knights lined up in a row occupying 20 distinct places as follows:
\begin{center}
\symknight\symknight\symknight\symknight\symknight\symknight\symknight\symknight\symknight\symknight
\symknight\symknight\symknight\symknight\symknight\symknight\symknight\symknight\symknight\symknight
\end{center}
Six of them will be replaced by rooks \symrook. How many possible replacements are there if no two of the rooks can be next to each other?
}


\textbf{Solution:}


\question{}{
Determine the number of ways to distribute 10 orange drinks, 1 lemon drink, and 1 strawberry drink to four thirsty students so that each student gets at least one drink, and the lemon and strawberry drinks go to different students.
}

\textbf{Solution:}



\question{}{
How many five-digit integers $n$  satisfy \textbf{all} of the following conditions:? (first digit cannot be $0$)\\
(a) $n>60000$.\\
(b) the digits are distinct.\\
(c) $n$ is even.\\
}

\textbf{Solution:}


\question{}{
There are $2n+1$ identical books to be put in a bookcase with three distinguishable shelves.
In how many ways can this be done if each pair of shelves together 
contains more books then the other? \\
\emph{(Only the number of books in each shelve gives a different way of putting. Substraction principle might help)}
}

\textbf{Solution:}


 
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