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\begin{center}
{\huge  \sc   Homework 8} \\
\smallskip
{ Due Date: 05/31/2022}\\
\smallskip
{Name:  }\\
{\large  }
\end{center}
\vspace{1cm}



\begin{enumerate}

\item[16.2] Liebeck, Einstein and Hawking pinch their jokes from a joke book which contains 12 jokes. Each year Liebeck tells six jokes, Einstein tells four and Hawking tells two( and everyone tells different jokes). For how many years can they go on, never telling the same three sets of jokes?
\newpage
\item[16.5]
\begin{enumerate}
    \item How many words of ten or fewer letters can be formed using the alphabet $\{a,b\}$
    
    \item Using the alphabet $\{a,b,c,d,e,f\}$, how many six letter words are there that use all six letters, in which no two of the letters $a,b,c$ occur consecutively?
\end{enumerate}
\newpage
\item[16.6]
\begin{enumerate}
    \item Find the number of arrangements of the set $\{1,2,\ldots, n\}$ in which the numbers $1,2$ appear as neighbours.
    
    \item Let $n\geq 5$. Find the number of arrangements of the set $\{1,2,\ldots, n\}$ in which the numbers $1,2,3$ appear as neighbours in order, and so do the numbers $4,5$.
\end{enumerate}
\newpage
\item[16.8]

\begin{enumerate}
    \item Prove that 
    \[
    {n+1 \choose r}={n \choose r} +{n \choose r-1} 
    \]
    
    \item Prove that for any positive integer $n$,
    \[
    3^n=\sum_{k=0}^n{n \choose k}2^k. 
    \]
\end{enumerate}
\newpage 
\item[16.9]
Give a proof of the Binomial Theorem 16.2 by induction on $n$. 

\newpage
\item[16.12]
The digits $1,2,3,4,5,6$ are written down in some order to form a six digit number.
\begin{enumerate}
    \item How many such six-digit numbers are there altogether?
    
    \item How many such numbers are even?
    
    \item How many are divisible by 4?
    
    \item How many are divisible by 8?
    
\end{enumerate}
\newpage

\item[16.13]

\begin{enumerate}
    \item Find the coefficient of $x^{15}$ in $(1+x)^{18}$.
    
    \item Find the coefficient of $x^{4}$ in $(2x^3-\frac{1}{x^2})^{8}$.
    
    \item Find the constant term in the expansion of  in $(y+x^2-\frac{1}{xy})^{10}$.
    
\end{enumerate}

\end{enumerate}




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