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{\bf MATH304  \hspace{1cm} HW 11\hfill \name}
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Due \textbf{Nov 19} before class, \textbf{answer without justification will receive 0 points}.
The typing the HW in \LaTeX{} is optional. 
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\question{}{
Determine the generating function for the number $h_n$ of
nonnegative integral solutions of
\[ 2e_1 + 5e_2 + e_3 + 7e_4 = n.\]
\emph{(Do not solve $h_n$, just create the generating function)}
}



\textbf{Solution:}



\question{}{
For every $n \geq 0$ determine the coefficient at $x^n$ in $(1 + x)^2(1 + x^2)^2(1 + x^4)^2(1 + x^8)^2\cdots$
which is equal to $\prod_{i=0}^{\infty}(1+x^{2^i})^2$.
}

\textbf{Solution:}


\question{}{
Find a(n ordinary) generating function where the coefficient of $x_k$ is the number of integers between 0 and $999,999$ whose sum of digits is $k$.
}

\textbf{Solution:}


\question{}{
Let $\alpha$ be a real number. Let the sequence $h_0,h_1,h_2,\ldots,h_n,\ldots$ be
defined by $h_0=1$, and $h_n = \alpha(\alpha-1)\cdots(\alpha-n+1),(n \geq 1)$.
Determine the exponential generating function for the sequence.
}

\textbf{Solution:}



\question{}{
\textbf{Using exponential generating function}, find the number of ways to put 30 labeled (thus distinct)
people into four different rooms $A, B, C, D$ if room $A$ must have an even number of people 
(possibly 0) and the other rooms must have at least one person.\\
\emph{(Solution without using exponential generating series will automatically get score 0.)}
}


\textbf{Solution:}

\question{}{
Find the number of $n$-digit numbers with digits 1,2,3, and 4 that have odd number of 1s and even number of 2s.
Use exponential generating functions.
}

\textbf{Solution:}






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