http://www.yorku.ca/nuri/econ3500/ch7-binomial-combinations-pg215.219.pdf
(relevant text 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)
permutations
permutations(3,2,repeats=TRUE)
permutations(3,2)
permutations(3,2,4:6)
x=4:6
permutations(3,2,x)
combinations
combinations(3,2)
combinations(3,2,1:3)
combinations(3,2,1:3,repeats=TRUE)
7.3
#
combinations and permutations
urnsamples(x,
size = ?, replace = ?, ordered = ?)
# x a vector or data frame from which sampling should take place.
# size number indicating the sample size.
# replace logical (TRUE/FALSE) whether sampling done with replacement.
# ordered logical (TRUE/FALSE) whether order is important.
# “In how many distinct ways can we permute the order of
our n distinct # objects?” p. 215
=====================================================================
Case1:
with replacement (repetitions allowed) and order matters
x=1:3
urnsamples(x,
size = 2, replace = TRUE, ordered = TRUE)
# n^2
expand.grid(x,x)
urnsamples(x,
size = 3, replace = TRUE, ordered = TRUE)
# n^n
expand.grid(x,x,x)
expand.grid(LETTERS[1:3],LETTERS[1:3])
# expand.grid(LETTERS[1:3],LETTERS[1:3],LETTERS[1:3])
=======================================================================
Case2
# without replacement (no repetitions) and order matters (PERMUTATIONS)
#
#
#
#
#
n ways of picking the first in line
(n - 1) ways of picking the second in line
(n - 2) ways of picking the third in line
. . .
1 way of picking the last object
# n(n - 1)(n - 2)(n - 3) . . . 1
# This product is called “n factorial” and is written “n!”; that is
# n! = n(n - 1)(n - 2)(n - 3) . . . 1
# In R
# factorial(n)
factorial(3)
# =6
factorial(4)
# = 24
x=c("Orange","Banana","Pear","Apple") # pages 215-216
urnsamples(x, size = 4, replace = FALSE, ordered = TRUE)
# 24 permutations
4!
# if with replacement (replace=TRUE) 4x4x4x4=256
permutations(4,4,x)
# Consider a clearer case: permutations of A, B, C
x=c("A","B","C")
# ch7 page 216
urnsamples(x, size = 3, replace = FALSE, ordered = TRUE)
#
permutations(3,3,x)
# x=LETTERS[1:3]
# permutations(3,3,LETTERS[1:3])
Page 217 permutation formula => n!/(n-r)!
Page 218:
x=c("A","B","C","D") # ch7 page 218
urnsamples(x, size = 2, replace = FALSE, ordered = TRUE)
# permutations(4,2,x)
#
#
#
#
“This list shows the permutations in which we are interested. List
the permutations in which we are not interested, the ones left out,
and count them. You should have (4! - 12) = 24 - 12 = 12
The 24 permutations where there are 2 containing AB (for example):
unique(urnsamples(x, size = 4, replace = FALSE, ordered = TRUE))
=======================================================================
Case3:
without replacement and order does not matter (COMBINATIONS)
page 218 indistinguishable objects
# n=5 r=2 (n-3)=3
n!/(n-r)!=20
x=c("A","A","A","B","C") # indistinguishable ch7 page 218
urnsamples(x, size = 5, replace = FALSE, ordered = TRUE)
unique(urnsamples(x, size = 5, replace = FALSE, ordered = TRUE))
# to see that there are 6 of each permutation
# y=urnsamples(x, size = 5, replace = FALSE, ordered = TRUE)
See: 7.3-binomial-combinations-pg218-duplicates.pdf
# permutations(5, 5, x, set=FALSE, repeats.allowed=FALSE)
# unique(permutations(5, 5, x, set=FALSE, repeats.allowed=FALSE))
# combinations => n!/r!(n-r)!
# page 219 n!/r!(n-r)! => 35
n=7, r=4 n-r=3
x=c("A","A","A","A","B","B","B") # indistinguishable
urnsamples(x, size = 4, replace = FALSE, ordered = FALSE)
urnsamples(1:7, size = 4, replace = FALSE, ordered = FALSE)
# combn(7,4)
# combinations(7,4)
# combinations(7,4,8:14)
The Binomial Expansions
“bi” means two.
means two terms.
Definition:
The factorial of an integer n ≥ 0, written n!, is
n × n-1 × ... × 2 × 1.
In particular, 0! = 1.
“four factorial”
“n factorial”
1!=1
The following is expression (7.1) on page 219
4!=1x2x3x4=24
In S-Plus the function
factorial
factorial(4)
[1] 24
factorial(0:4)
[1] 1 1 2 6 24
combinations’ expression (7.1) on pg 219 is
the function choose
choose(n,x)
choose(4,2)
[1] 6
choose(4,0:4)
In S-Plus ‘
[1] 1 4 6 4 1
The ‘permutations’ expression on pg 218
[n!/(n-r)!] is the function
Choose(n,r,order=T)
choose(4,2,order=T)
[1] 12
Pascal’s Triangle (pgs 227 and 591-592 of text)
Expand
Expression
=
Expression
=
Expression
(a + b)4 = a4 + 4a3b + 6a2b2 + 4ab3 + b4
Expand
Pascal’s Triangle (pg 591 of text) is a triangle of coefficients that is based on the
binomial theorem.
It is formed by adding the two numbers directly above and placing 1’s on the outer sides.
1
1
1
1
1
1
1
1
1
The binomial
7
1
8
9
36
2
3
4
5
6
21
84
1
3
6
10
15
28
1
1
4
10
20
35
56
126
5
15
35
70
1
1
6
21
56
126
7
28
84
coefficient expansion is as follows:
Expand the following:
1
1
8
1
36
9
1
Expand
Expression
Result
In general,
Expression
Result
Expression
Result
(relevant text pages on combinations)
http://www.yorku.ca/nuri/econ3500/ch7-binomial-combinations-pg215.219.pdf