REVIEW OF COUNTING
---------Warren D Smith April 2003----------------


Pick r animals from n.
Total number of possible r-element SETS you can get in this way is   
          n!
nCr = --------- = "r choose r"
      (n-r)! r!
orders do NOT matter. Same set if same animals, even if lined up in other order.
  AB and BA  do NOT count as 2 different sets.



Pick r baseball players from n.
Total number of possible r-man baseball lineups you can get in this way is   
        n!
nPr = ------
      (n-r)!
orders DO matter. AB counts as different lineup than BA.
nPr  is  BIGGER than nCr  (if r>1).




# L-letter words, each letter selected from an alphabet of size A:
        L
       A  .    (BIGGER than  APL  if r>1.)



# orderings of n things:   n! = 1*2*3*4*...*(n-1)*n.     
         0!=1, 1!=1, 2!=2, 3!=6, 4!=24, 5!=120.



EXAMPLES:
            5!      5*4*3*2*1    5*4   
   5C3 = ------- = ----------- = --- = 10   ways to pick 3-animal sets from 5 animals.
          3!  2!   3*2*1 * 2*1    2            namely ABC ABD ABE ACD ACE
                                                      ADE BCD BCE BDE CDE  from ABCDE.

   3! = 6 = 3*2*1    ways to order 3 things, namely  ABC ACB CAB CBA BAC BCA.

           5!
   5P3 = ----- = 5*4*3 = 60          ways to pick 3-man lineups from 5.
           2!                            note 6 times bigger than 5C3 since 6 orderings of each 3-animal set
                                                all now count as different.
    3
   5  = 125                         3-letter words, each letter from a 5-letter alphabet.
                                        note BIGGER than 5P3 since now allowed to RE-USE letters (or baseball players)
                                        giving you more possibilities.

   5! = 120                      ways to order 5 things.

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