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\begin{document}


{\bf MATH304  \hspace{1cm} HW 8\hfill \name}
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\begin{minipage}[t]{0.75\textwidth}
Due \textbf{Oct 22} before class, \textbf{answer without justification will receive 0 points}.
The solution has to be typed (using \LaTeX). 
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\question{}{
If you pick an integer between 1 and 1000 (including 1 and 1000), what is the probability that it is either divisible by 7 or 5 or even (or two or all of these)?
}

\textbf{Solution:}


\question{}{
How many multisets of 3 letters can be formed from letters M,I,S,S,I,S,S,I,P,P,I?
}

\textbf{Solution:}


\question{}{
Count the number of integer solutions of
\[
 x_1+x_2+x_3+x_4 = 28,
\]
where $0 \leq x_1 \leq 6$, $0 \leq x_2 \leq 10$, $0 \leq x_3 \leq 15$, $0 \leq x_4 \leq 21$.
}

\textbf{Solution:}



\question{}{
How many ways are there to distribute $k$ distinct objects into five (distinct) boxes with at least
one empty box?
}

\textbf{Solution:}


\question{}{
Count the number of placements of 8 tokens on $4 \times 4$ board such that there
exists a row or a column containing 4 tokens.
}

\textbf{Solution:}


\question{}{
Prove that $D_n$ is an odd number if and only if $n$ is an even number.
}

\textbf{Solution:}


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