# 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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