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Math 491, Problem Set \#9 \\
(due 10/9/03; postponed to 10/14/03)
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\item
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\item[(a)]
How many different polygonal paths of length $n$ are there
that start at the point $(0,0)$
and then take $n$ steps of length 1,
such that each step is either rightward, leftward, or upward,
and such that no point gets visited more than once?
Give an explicit formula.
% Next time, explain what a polygonal path is!
\item[(b)]
If one chooses at random
one of the paths of length $n$ described in part (a)
(so that each of the length-$n$ paths has an equal chance of being chosen),
what is the expected value of the $y$-coordinate of the last point on the path?
Find a constant $c$ so that
this expected value is asymptotic to $cn$.
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