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\lhead{MATH 387-01}
\chead{Problem Set \#1\answer{ solutions}}
\rhead{\qonly{Name: \framebox{\phantom{Your name goes here}}}}
\lfoot{}
\cfoot{Page \thepage\ of \pageref{LastPage}}
\rfoot{{\bf due September 4, 2013}}
\begin{document}
\qonly{Show all your work, and explain why you use the arithmetic
operations you use in reaching an answer.}
\begin{enumerate}
\item {\bf (10 points)} \question{Let us call a number ``good'' if all
of its digits are the same parity (i.e. all odd or all even). So,
for instance, 62880 and 31713 are both good, but 61407 is not,
since 6, 4, and 0 have different parities from 1 and 7.}
\begin{enumerate}
\item {\bf (5 points)} \question{Determine the number of 3-digit
``good'' numbers; note that numbers, when conventionally
written, do not have a leading 0; e.g. ``024'' is not a
three-digit number.}
\answer{}
\item {\bf (5 points)} \question{Find a formula in terms of $n$ for
the number of $n$-digit ``good'' numbers.}
\answer{}
\end{enumerate}
\item {\bf (10 points)} \question{A modified deck of cards contains
five suits and ten cards (numbered 1--10) in each suit. A
``straight'' is defined as in poker as a hand in which the five
cards are numerically consecutive, and of any suit (ignore, for
the purposes of this problem, the poker convention that a hand of
this sort could also be a ``straight flush'' or ``royal flush''
instead). How many different five-card hands (which are not
ordered) are straights?}
\answer{}
\item {\bf (10 points)} \question{Let us consider license plates which
consist of two consecutive groups of either three letters (from a
pool of 26 possible) or three numbers (from a pool of 10
possible); e.g. 134-WRG, EEE-EEK, and 852-584 are all possible
license plates. However, to avoid confusion, you forbid either
group of 3 to consist entirely of the numbers ``0'', ``1'', and
``5'', or the letters ``O'', ``I'', and ``S'' (so we don't allow
115-RTG, because it might be misread as IIS-RTG). How many license
plates are possible according to this scheme (you may leave the
answer as an unsimplified arithmetic expression, if you like).}
\answer{}
\item {\bf (10 points)} \question{Suppose we build a random
four-letter ``word'' by placing in order a random selection from
the 21 consonants, a random selection from the 5 vowels, another
random consonant, and then another random vowel (most words we
produce this way will be nonsense words like ``LEFE''). What is
the probability that we make a word in which all four letters are
different, like ``XOFA''?}
\answer{}
\end{enumerate}
\vfill
\begin{center}
\fbox{
\begin{minipage}{6in}
Musica est exercitium arithmeticae occultum nescientis se numerare animi.
\hfill ---Gottfried Leibniz
\end{minipage}
}
\end{center}
\end{document}