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The Choose Numbers Some Sample Choose Numbers Some Sample Choose

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					                                                                      Some Sample Choose Numbe rs
             The Choose Numbers
                                                             P What is 4 choose 1?
P How many w ays are ther e to s elect k objects fr om         – 4
  n d ist inct objects ?                                      < What is 4 choose 2?
                                                               – 6
 < We called t hat nu mber “ n choose k ”
                                                              < What is 5 choose 1?
 < We denoted it  
                    n
                    
                    k                                        – 5
P I’ll informa lly call s uc h numbers the “c hoos e          < What is 5 choose 2?
  numbers”                                                     – 10
                                                              < What is 5 choose 3?
                                                               – 10
                                                               – Not e th at t hi s i s th e s am e as 5 choos e 2.
                                                               – That’ s becaus e s el ectin g which t hree to pi ck is t he sam e as
                                                                 s el e ctin g whi ch two not to pi ck
                                                               – And t hus th ere are th e sam e numb ers of ways t o m ake the
                                                                 s el e ctio ns




       Some Sample Choose Numbers
                                                                                 Choose Polynomials
P What is 5 choos e 4
  < Same as 5 choose 1, which is 5                            PFor fixed k, the value of n c hoose k is a
PIn gener al,  n =  n 
                                                           polynomial in n of degree k
               k  n − k 
PIn gener al, what is n choose 1?                             PFor example:
  <n                                                             n
                                                                  =1
P What is n choose n?                                            0
  <1                                                             n
                                                                  =n
P What is n choose 0?                                            1
  <1                                                             n
                                                                  = (1 / 2) n 2 − ( 1 / 2) n
P What is n choose 2?                                            2
  < n(n – 1) / 2, which can be derived from the fact orial       n
                                                                  = (1 / 6) n3 − (1 / 2 )n 2 + (1 / 3) n
    expression for choose numbers                                3
                         Pascal’s Triangle
                                                P as cal’s Tri angl e is
                                                                                               Binomial Coefficie nts
 T his t ri angl e cont ain s all o f the        de fi ned recursiv el y:
 choos e num bers.
                                                                                  PThe res t of the w orld calls the choos e numbers
                                                    T here are all 1's on           “binomial c oeffic ients ”
 T o fi nd n choos e k, go to t he
                                                      th e out si de, and
 nt h row and s el ect th e kt h
 ent ry.                                                i nsi de, every           PThat is because they appear in the expans ion of
                                                          ent ry is th e            binom ials to integer pow ers
                                                            s um o f th e
 B ut bear in min d t hat
 P as cal’s t ri angl e start s
                                                               t wo ent ri es     P1
                                                                  above it.
 wit h t he 0t h row, and                                                         Pa + b
 each row st art s with
 t he 0t h entry.                                                                 P a 2 + 2ab + b 2
                                                                                  P...




      Binomial Coefficients are Choose
                 Numbers                                                                              Another Question
                                (a + b)5                                          P How many w ays ar e ther e to place n balls into
                                   =                                               two bins, one bin blue and the other bin red, s o
                 (a + b)·(a + b)·(a + b)·(a + b)·(a + b)                           that exactly k balls go into the blue bin and n – k
                                                                                   go into the red bin?
In the first expression, t he coefficient of a2 b3, for exa mp le,
is w hat w e call a binomial coefficient.                                                                                  n balls

T o comp ut e t he coefficient of a2 b3 in the second expression,
                                                                                      k balls int o                       n – k balls int o
w e count the number of w ays t o select a t erm from each of
the five factors, making sure t hat exactly 3 of them are “ b”.                           this bin                        this bin
T here are 5 choose 3 w ays t o do t hat.
                                                                                < The answer is n choose k, since w e can simp ly count the
                                                                                  number of w ays to s elect t he k for t he blue bin.
                               More Bins                                            Multinomial Coefficients
  P How many w ays are ther e to plac e n d ist inct
    objec ts into t d ist inct bins s o that                            bins
    < the number of objects in each bin is k 1, k 2, ..., k t        objec ts   1    2   3   4   5   6   7   8   9 10 11 12 13 14 15 16
    < w here k 1 + k 2 + þ+ k t = n?
  P Can you th ink of a “c hoos e number ” w ay to do                 P Every rear rangement of the bins g ives another
    this?                                                               ass ignment of objects to bins.
  P Here is an anagram way to do this:                                P The number of “bin anagrams” is 16! / 5!5!3!3!
   bins                                                               P In gener al, there ar e n! / k 1!k 2!... k t!
objec ts   1   2   3   4   5    6   7   8   9 10 11 12 13 14 15 16    P Thes e number are called mult inomial coeff ic ients
  P Ever y rearrangement of the bins gives another
    ass ignment of objects to bins.




               Some Sample Proble ms
  P How many w ays are ther e to divide tw enty
   students into four teams, the A, B, C and D
   teams, each c ontain ing five students?
    < 20! / 5!5!5!5! = 11,732,745,024
  PSix s tudents wer e elected as off ic ers, and the
   teac her w ishes to ass ign one of those students as
   pres id ent, tw o as vic e pres idents and three as
   secretar ies. How many was ar e ther e to do this?
    < 6!/1!2!3! = 60

				
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