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http://www.yorku.ca/nuri/econ3500/ch7-binomial-combinations-pg215.219.pd  

"rele#ant te$t pages%

Ch7

install.packages("combinat")library(combinat)

install.packages("gtools")

library(gtools)

install.packages("prob")

library(prob)

library(combinat)

library(gtools)library(prob)

# combinations {gtools}

combinations(3,2,letters[1:3])

combinations(3,2,letters[1:3],repeats=TRUE)

permutations(3,2,letters[1:3])

permutations(3,2,letters[1:3],repeats=TRUE)

permutationspermutations(3,2,repeats=TRUE)

permutations(3,2)

permutations(3,2,:!)

"=:!

permutations(3,2,")

combinations

combinations(3,2)combinations(3,2,1:3)

combinations(3,2,1:3,repeats=TRUE)

7.3 combinations an$ permutations

% urnsamples(", si&e = ', replace = ', or$ere$ = ')  

% " a ector or $ata rame rom *hich samplin+ shoul$ tae place-

http://www.yorku.ca/nuri/econ3500/ch7-binomial-combinations-pg215.219.pdfhttp://www.yorku.ca/nuri/econ3500/ch7-binomial-combinations-pg215.219.pdf

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% si&e number in$icatin+ the sample si&e-

% replace lo+ical (TRUE./0E) *hether samplin+ $one *ith replacement-

% or$ere$ lo+ical (TRUE./0E) *hether or$er is important-

# “In how many distinct ways can we permute the order of ourn distinct # objects?” p. 21

!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

"ase1:

 with replacement (repetitions allowed ) an$ order matters

!1$3

urnsamples%& si'e ! 2& replace ! ()*+& ordered ! ()*+,  & n'2

e$pand.grid"$($%

urnsamples(", si&e = 3, replace = TRUE, or$ere$ = TRUE)  & n'n

e$pand.grid"$($($%

e$pand.grid")*++*,1:3()*++*,1:3%

& e$pand.grid")*++*,1:3()*++*,1:3()*++*,1:3%

!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

"ase2 

% *ithout replacement (no repetitions) an$ or$er matters %-+)*(/(I0,

% n *as o picin+ the irst in line

% (n 4 1) *as o picin+ the secon$ in line

% (n 4 2) *as o picin+ the thir$ in line

% - - -

% 1 *a o picin+ the last ob5ect

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% n(n 4 1)(n 4 2)(n 4 3) - - - 1

% This pro$uct is calle$ 6n actorial an$ is *ritten 6n89 that is

% n8 = n(n 4 1)(n 4 2)(n 4 3) - - - 1

% n R

% actorial(n)

factorial%3,  % =!

factorial%,  % = 2

!c%40ran5e4&46anana4&4-ear4&4/pple4, % pa5es 21218

urnsamples%& si'e ! & replace ! 9/:+& ordered ! ()*+, 

% 2 permutations 8

% i *ith replacement (replace=TRUE) """=2;!

 permutations%&&,

# "onsider a clearer case$ permutations of /& 6& "

"=c(

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uniue%urnsamples%& si'e ! & replace ! 9/:+& ordered ! ()*+,,

=======================================================================

"ase3:

*ithout replacement an$ or$er $oes not matter %"06I/(I0,

 pa5e 21> in$istin+uishable ob5ects

% n=; r=2 (n43)=3 n8.(n4r)8=2A

"=c(

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Definition

!e factorial of an integer n $, written n%, is

n & n' & ... & 2 & .

n partic*lar, $% + .

“fo*r factorial”

“n factorial”

%+

!e following is epression (-.) on page 2

4!=1x2x3x4=24

In S-Plus the function factorial

factorial(4)[1] 24factorial(0:4)[1] 1 1 2 24

In S-Plus co"#inations$ ex%ression (&'1) on % 21 isthe function choose

choose(n*x)choose(4*2)[1] choose(4*0:4)

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[1] 1 4 4 1

+he %er"utations$ ex%ression on % 21, 

[n!(n-r)!] is the function.hoose(n*r*or/er=+)choose(4*2*or/er=+)[1] 12

ascals +riangle "pgs 227 and 591-592 o te$t%

*$pand

Expression

+

Expression

+

Expression

(a #)4 = a4  4a3# a2#2  4a#3  #4

/pan0

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ascals +riangle ( pg 1 of tet) is a triangle of coefficients tat is base0 on te binomial teorem.

t is forme0 by a00ing te two n*mbers 0irectly aboe an0 placing s on te o*ter si0es.

2

3 3

4 5 4

1 $ $ 1

5 1 2$ 1 5

- 2 31 31 2 -

6 26 15 -$ 15 26 6

35 64 25 25 64 35

!e binomial coefficient epansion is as follows

/pan0 te following

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Expand

Expression

Result

In general,

Expression

Result

Expression

Result

"rele#ant te$t pages on combinations%

http://www.yorku.ca/nuri/econ3500/ch7-binomial-combinations-pg215.219.pd  

http://www.yorku.ca/nuri/econ3500/ch7-binomial-combinations-pg215.219.pdfhttp://www.yorku.ca/nuri/econ3500/ch7-binomial-combinations-pg215.219.pdf

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