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\centerline{Mathematics 357 -- Combinatorics}
\centerline{Review Sheet for Midterm Exam 1}
\centerline{February 18, 2005}
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{\it General Information}
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As announced in the course syllabus and on the 
course homepage, the first midterm exam will be given
in class on Friday, February 25.  The exam will cover
the material we have studied since the start of the
semester, through the material on generalizations
of the binomial theorem that we will discuss 
on Friday, February 18. (This includes sections from Chapters 
2,3,4,5 in the text -- see detailed list of topics you are 
responsible for below).  This will be an individual, closed-book, 
closed-notes exam.  No collaboration of any kind will be permitted
during the exam period.  You {\it may} bring a calculator
if you want one; however, answers written terms of powers, 
factorials, binomial coefficients, etc.  can always be 
left in that form, so a calculator should be of truly 
minimal value for your work on the exam.  My advice: 
leave it home!

If there is interest, I would be happy to run a late
afternoon or evening review session next week to help
you prepare.  Schedule constraints:  I will have to leave
campus no later than 5:15pm on Thursday and we are giving
an evening exam in MATH 132 on Wednesday evening, so I
will be busy from 6:00pm to 9:00pm or so that day.  In addition,
I will not be available Monday evening after 5:30pm or so.

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{\it Topics to be Covered}
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\item{1)}  Chapter 2:  The Pigeonhole Principle (basic and strong forms), 
applications.
\item{2)}  Chapter 3:  Basic counting principles, permutations and
combinations of sets, permutations and combinations of 
multisets.  (Note: some counting problems may come from 
considerations of ``sets'' and other structures in the 
Set card game.  Review Discussion 1 to make sure you recall
how all that works!)
\item{3)}  Chapter 4:  Generating combinations.  (Note -- we
only discussed a portion of the material in this chapter, and
you're only responsible for the stuff from sections 3 and 4 that
we did cover.)
\item{5)}  Chapter 5:  Binomial coefficients, identities, algebraic
and combinatorial proofs, 
applications to counting problems.  (Omit the material
in sections 4 and 7 of this chapter.)

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Note:  Some problems may ask you to find algebraic and/or combinatorial
arguments to prove identities concerning binomial or multinomial
coefficients, etc.  The best way to prepare for these is to 
study the examples we did in class and make sure you understand
the basic ideas.

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{\it Some Good Practice Problems}
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\item{$\bullet$} Chapter 2/3, 16, 17, 19, 27
\item{$\bullet$} Chapter 3/11, 12, 14, 23, 26, 37, 39 
\item{$\bullet$} In the card game Set, the cognoscenti sometimes call ``sets''
where the three cards are all different in each of the four
attributes ``null sets''.  How many different ``null sets''
are there in the Set deck?  
\item{$\bullet$} Chapter 4/11, 12, 15, 16, 18 (see Discussion 2!)
\item{$\bullet$} Chapter 5/11, 18, 19, 20, 23, 29, 30, 38, 39, 44.
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Warning:  The above problems cover the ideas you will need to
understand in order to do the exam problems, but the exam
problems may be organized and formatted differently, may
draw on different types of examples, etc.  Be prepared!
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