1 \(n\) Factorial in R

\(n! = n \times (n-1) \times ... \times 2 \times 1\) is represented by factorial(n).

Examples:

factorial(3)

[1] 6

factorial(6)

[1] 720

2 Combination in R

Calculate the number of ways to choose \(k\) items from \(n\) items, \(^nC_k = \frac{n!}{k!(n-k)!}\) is represented by choose(n, k).

Examples:

# Number of ways to choose 2 items from 5 items
choose(5, 2)

[1] 10

# Number of ways to choose 5 items from 10 items
choose(10, 5)

[1] 252

To list all the ways to choose \(k\) items from \(n\) items, use the combn() function.

Examples:

# List of ways to choose 2 items from 5 items
combn(1:5, 2)

     [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10]
[1,]    1    1    1    1    2    2    2    3    3     4
[2,]    2    3    4    5    3    4    5    4    5     5

# List of ways to choose 2 letters from A to D
combn(c("A", "B", "C", "D"), 2)

     [,1] [,2] [,3] [,4] [,5] [,6]
[1,] "A"  "A"  "A"  "B"  "B"  "C" 
[2,] "B"  "C"  "D"  "C"  "D"  "D" 

# List of ways to choose 3 letters from A to D
combn(c("A", "B", "C", "D"), 3)

     [,1] [,2] [,3] [,4]
[1,] "A"  "A"  "A"  "B" 
[2,] "B"  "B"  "C"  "C" 
[3,] "C"  "D"  "D"  "D" 

3 Permutation in R

Calculate the number of ways to arrange \(n\) different items, which is \(n!\).

Examples:

# Number of ways to arrange 4 different items
factorial(4)

[1] 24

# Number of ways to arrange 10 different items
factorial(10)

[1] 3628800

To list all the ways to arrange \(n\) different items, use the permn() function after installing the "combinat" package.

Examples:

# Install the package
install.packages("combinat")
# Load the package
library(combinat)

# List all the ways to arrange 3 different items
permn(3)

[[1]]
[1] 1 2 3

[[2]]
[1] 1 3 2

[[3]]
[1] 3 1 2

[[4]]
[1] 3 2 1

[[5]]
[1] 2 3 1

[[6]]
[1] 2 1 3

# List all the ways to arrange A to C
permn(c("A", "B", "C"))

[[1]]
[1] "A" "B" "C"

[[2]]
[1] "A" "C" "B"

[[3]]
[1] "C" "A" "B"

[[4]]
[1] "C" "B" "A"

[[5]]
[1] "B" "C" "A"

[[6]]
[1] "B" "A" "C"