Date: Sat, 3 Nov 2007 13:18:23 -0600 (CST)
   From: Mark Jacobson <jacobson@math-cs.cns.uni.edu>
     To: 810-080-02-fall@uni.edu
Subject: C(n,r) and P(n,r) and ITT 307 office and assignment...

Hi 080 students,

      The Midterm Test two will be on November 15th, Thursday.

      The week number 12 row has a link to PART of the assignment that
is due on November 6th.                   ----

      http://cns2.uni.edu/~jacobson/c080.html

      http://cns2.uni.edu/~jacobson/121/assign080.txt

      Copies of the 11/06 due homework are taped up outside my office
door, and I just put some more up there.  The building is open today
(Saturday afternoon), as of right now.

      You should have gotten 14 out of 14 possible on the online quiz by
now.  It is interactive, so you will know what score you have whenever
you want via the Grade Quiz button.

      http://cns2.uni.edu/~jacobson/js/framesDiscrete080.html

      page 399 of the textbook has the C(n, r) definition.

      How many different committees of size 4 are possible for our
      discrete class of 29 students?

      C(29, 4) is read 29 choose 4 and a committee of 4 students would
               be a subset of size 4 of 810:080 section 02 students.

      It is also called a 4-combination (see page 399).

      29*28*27*26
     ------------- = 23,751 different 4 person committees are possible.
        4*3*2*1

     See Definition and Theorem 2 on page 393 for PERMUTATION concept and
     formula.

           P(n, r) = n * (n-1) * n-2) * ... * (n-r + 1)

      It can also be expressed as:
                        n!
           P(n, r) = ----------
                      (n - r)!

      So how many different ways could our class of 29 students have 4
      different officers:
                                  President
                             Vice President
                                  Secretary
                                  Treasurer

      P(29, 4) = 29 * 28 * 27 * 26 ways to have the officers

                     29!        29*28*27*26*25!
      P(29, 4) = ----------  = ---------------- = 29*28*27*26 = 570,024
                  (29 - 4)!          25!

      How many ways can the Gold, Silver, Bronze medals be awarded in
      an Olympic event that has 20 competitors?

      20 * 19 * 18 = P(20, 3) = 6,840 is the answer.

      How many different Olympic teams are possible if the qualifying race
      has 20 runners and the top three runners will represent the country?

      C(20, 3) = 20*19*18   20*19*18
                 -------- = -------- = 20 * 19 * 3 = 60 * 19 = 1,140
                    3!         6

      Note that n! reads as n factorial.

      Factorial is explained on pages 15 and 16 of the book.

      n! = n * (n-1) * (n-2) * ... * 3 * 2 * 1

      1! = 0! = 1

      How many ways can the 3 letters U N I be arranged as a sequence, i.e.
      how many 3 letter words can be made with U, N and I?

      3 factorial words, which is 3 * 2 * 1 = 6 words.

      3 choices for the first letter,
      2 choices left for what to use as the 2nd letter, and
      1 letter left to write down as the 3rd and last letter of the word

      U           or         N         or         I           3 choices
   N  or  I              U   or   I            U  or  N       2 choices

   I      N              I        U            N      U       1 choice
  ---    ---            ---      ---          ---    ---
  UNI    UIN            NUI      NIU          IUN    INU  six different
                                                          words, aka
                                                          permutations of
                                                          "U", "N", "I"

      Mark