Date: Tue, 17 Oct 2006 11:33:12 -0500 (CDT)
   From: Mark Jacobson <jacobson@math-cs.cns.uni.edu>
     To: 810-080-02-FALL@uni.edu
Subject: Test on Friday...

Hello Discrete students,

There are 12 presents under the tree. Gollum is told he can pick any 1 or 
any 2 or any 3 presents? How many different possibilities exist for what 
Gollum can have under these circumstances?

Question number 3 at
     http://www.cns.uni.edu/~jacobson/js/framesDiscrete080.html

     is a great way to practice the counting problems involving
     P(n,r) and C(n,r) and the rule of product and rule of sum.

http://www.cns.uni.edu/~jacobson/080/email080quiz3.txt

      11 men and 8 women problem, explained by using
      a more reasonable problem size of 4 cars and 3 trucks.

      No two women can sit together (next to each other).

      No two trucks can be parked next to each other.  They
      must always have at least one car in between them and the
      next truck.

      http://www.cns.uni.edu/~jacobson/080/email080quiz3.txt

Can you prove the following easily without looking at the solution?

     Prove that any amount of postage >= 64 cents can be built
     using only 5-cent and 17-cent stamps.

     Does your proof include all the necessary steps?

     Did you CLEARLY indicate the IH (inductive hypothesis) THREE times?
                                      I         H

     Did you write down the multiples of 5 and 17 next to each other
     and break it down into the correct two cases?

     Practice until you get it perfect.  Repetition in mathematics is
     very, very, very important.

     Practice until you uinderstand it deeply.


Can you prove the following for all n >= 1?

                                        5n(n + 1)
Prove that 5 + 10 + 15 + ... + 5n = ----------------
                                           2

     Do it and then look at the following page to compare your result
     to the solution shown there.

     What parts did you miss or do you need to improve on?

     Did you have trouble with the algebra?  Guess what.  Your algebra
     will improve greatly the more times you review this and other proofs.

     Did you have trouble with the pattern of inductive proofs?

     Basis Step:   Compare it to the solution shown.

     Inductive Step:   Did you set it up correctly????

     I.  Assume as the IH that for some n = k >= 1, that

            ....

    II.  Try to prove the GOAL for n = k + 1 that



   III.  Proof begins here for inductive step,

            and somewhere you use and you state that you are using
            the IH...

      http://www.cns.uni.edu/~jacobson/080/inductionOct2003.txt

      Study the above solution and compare to your solution.

      Solve the problem again tommorrow or later the same day.

      Does it go faster?   Do you have a solution that is closer to
                           perfect and closer to getting full credit?

      Does the algebra go faster and make more sense when you do it
      again a 2nd time or a 3rd time?

      Test two on Friday.

      http://www.cns.uni.edu/~jacobson/inductive.html

      Practice, practice, practice.

      http://www.cns.uni.edu/~jacobson/c080.html

Mark