Introduction to Probability, Grinstead, C. Snell, J., 1997
Page 115
Exercise 21

Question
A lady wishes to color her fingernails on one hand using at most two of the colors red, yellow, and blue. How many ways can she do this?

Answer
There are two possible scenarios:
Scenerio 1.) The lady colored her finger using only one color. This scenerio represents three distinct solutions since there are three colors.
\(C(n,k)= n!/k!(n−k)!\)
\(C(3,1)= 3!/1!(3−1)!\)
\(C(3,1)= (3*2*1)/(1*2*1)\)
\(C(3,2)=3\)

Scenerio 2.) The lady uses exactly two colors.

\(2^5=32\) There are 32 possible permutations. However, two of these permutations necessarily have the lady as only painting one color on all five fingers. Therefore, we subtract two.
\(32-2=30\) Subtract two because two scenarios where the color would be all one of the two colors.

30*3 different color permutations (red-yellow, red-blue, yellow-blue) = 90 different permutations

scenerio 1 + scenario 2= 3 + 90 = 93 different ways.

The below code validates that there are 93 different ways for the lady to paint her finger nails.

#install.packages("gtools")
library(gtools)
q=c('r', 'y','b')
x <- c('r', 'y')
y <- c('r', 'b')
z <- c('y', 'b')

blue_red    <- permutations(n=2,r=5,v=y,repeats.allowed=T)   
blue_yellow <- permutations(n=2,r=5,v=z,repeats.allowed=T)
red_yellow  <- permutations(n=2,r=5,v=x,repeats.allowed=T)
total_permutations <- (nrow(blue_red) + nrow(blue_yellow) + nrow(red_yellow))-6  
i <- nrow(combinations(n=3, r=1, v=q, repeats.allowed=F)) # three choose one scenario 1 


answer<-total_permutations+i

answer # this the number of ways the lady could have painted her fingers
## [1] 93
print(blue_red )
##       [,1] [,2] [,3] [,4] [,5]
##  [1,] "b"  "b"  "b"  "b"  "b" 
##  [2,] "b"  "b"  "b"  "b"  "r" 
##  [3,] "b"  "b"  "b"  "r"  "b" 
##  [4,] "b"  "b"  "b"  "r"  "r" 
##  [5,] "b"  "b"  "r"  "b"  "b" 
##  [6,] "b"  "b"  "r"  "b"  "r" 
##  [7,] "b"  "b"  "r"  "r"  "b" 
##  [8,] "b"  "b"  "r"  "r"  "r" 
##  [9,] "b"  "r"  "b"  "b"  "b" 
## [10,] "b"  "r"  "b"  "b"  "r" 
## [11,] "b"  "r"  "b"  "r"  "b" 
## [12,] "b"  "r"  "b"  "r"  "r" 
## [13,] "b"  "r"  "r"  "b"  "b" 
## [14,] "b"  "r"  "r"  "b"  "r" 
## [15,] "b"  "r"  "r"  "r"  "b" 
## [16,] "b"  "r"  "r"  "r"  "r" 
## [17,] "r"  "b"  "b"  "b"  "b" 
## [18,] "r"  "b"  "b"  "b"  "r" 
## [19,] "r"  "b"  "b"  "r"  "b" 
## [20,] "r"  "b"  "b"  "r"  "r" 
## [21,] "r"  "b"  "r"  "b"  "b" 
## [22,] "r"  "b"  "r"  "b"  "r" 
## [23,] "r"  "b"  "r"  "r"  "b" 
## [24,] "r"  "b"  "r"  "r"  "r" 
## [25,] "r"  "r"  "b"  "b"  "b" 
## [26,] "r"  "r"  "b"  "b"  "r" 
## [27,] "r"  "r"  "b"  "r"  "b" 
## [28,] "r"  "r"  "b"  "r"  "r" 
## [29,] "r"  "r"  "r"  "b"  "b" 
## [30,] "r"  "r"  "r"  "b"  "r" 
## [31,] "r"  "r"  "r"  "r"  "b" 
## [32,] "r"  "r"  "r"  "r"  "r"
print(blue_yellow)
##       [,1] [,2] [,3] [,4] [,5]
##  [1,] "b"  "b"  "b"  "b"  "b" 
##  [2,] "b"  "b"  "b"  "b"  "y" 
##  [3,] "b"  "b"  "b"  "y"  "b" 
##  [4,] "b"  "b"  "b"  "y"  "y" 
##  [5,] "b"  "b"  "y"  "b"  "b" 
##  [6,] "b"  "b"  "y"  "b"  "y" 
##  [7,] "b"  "b"  "y"  "y"  "b" 
##  [8,] "b"  "b"  "y"  "y"  "y" 
##  [9,] "b"  "y"  "b"  "b"  "b" 
## [10,] "b"  "y"  "b"  "b"  "y" 
## [11,] "b"  "y"  "b"  "y"  "b" 
## [12,] "b"  "y"  "b"  "y"  "y" 
## [13,] "b"  "y"  "y"  "b"  "b" 
## [14,] "b"  "y"  "y"  "b"  "y" 
## [15,] "b"  "y"  "y"  "y"  "b" 
## [16,] "b"  "y"  "y"  "y"  "y" 
## [17,] "y"  "b"  "b"  "b"  "b" 
## [18,] "y"  "b"  "b"  "b"  "y" 
## [19,] "y"  "b"  "b"  "y"  "b" 
## [20,] "y"  "b"  "b"  "y"  "y" 
## [21,] "y"  "b"  "y"  "b"  "b" 
## [22,] "y"  "b"  "y"  "b"  "y" 
## [23,] "y"  "b"  "y"  "y"  "b" 
## [24,] "y"  "b"  "y"  "y"  "y" 
## [25,] "y"  "y"  "b"  "b"  "b" 
## [26,] "y"  "y"  "b"  "b"  "y" 
## [27,] "y"  "y"  "b"  "y"  "b" 
## [28,] "y"  "y"  "b"  "y"  "y" 
## [29,] "y"  "y"  "y"  "b"  "b" 
## [30,] "y"  "y"  "y"  "b"  "y" 
## [31,] "y"  "y"  "y"  "y"  "b" 
## [32,] "y"  "y"  "y"  "y"  "y"
print(red_yellow)
##       [,1] [,2] [,3] [,4] [,5]
##  [1,] "r"  "r"  "r"  "r"  "r" 
##  [2,] "r"  "r"  "r"  "r"  "y" 
##  [3,] "r"  "r"  "r"  "y"  "r" 
##  [4,] "r"  "r"  "r"  "y"  "y" 
##  [5,] "r"  "r"  "y"  "r"  "r" 
##  [6,] "r"  "r"  "y"  "r"  "y" 
##  [7,] "r"  "r"  "y"  "y"  "r" 
##  [8,] "r"  "r"  "y"  "y"  "y" 
##  [9,] "r"  "y"  "r"  "r"  "r" 
## [10,] "r"  "y"  "r"  "r"  "y" 
## [11,] "r"  "y"  "r"  "y"  "r" 
## [12,] "r"  "y"  "r"  "y"  "y" 
## [13,] "r"  "y"  "y"  "r"  "r" 
## [14,] "r"  "y"  "y"  "r"  "y" 
## [15,] "r"  "y"  "y"  "y"  "r" 
## [16,] "r"  "y"  "y"  "y"  "y" 
## [17,] "y"  "r"  "r"  "r"  "r" 
## [18,] "y"  "r"  "r"  "r"  "y" 
## [19,] "y"  "r"  "r"  "y"  "r" 
## [20,] "y"  "r"  "r"  "y"  "y" 
## [21,] "y"  "r"  "y"  "r"  "r" 
## [22,] "y"  "r"  "y"  "r"  "y" 
## [23,] "y"  "r"  "y"  "y"  "r" 
## [24,] "y"  "r"  "y"  "y"  "y" 
## [25,] "y"  "y"  "r"  "r"  "r" 
## [26,] "y"  "y"  "r"  "r"  "y" 
## [27,] "y"  "y"  "r"  "y"  "r" 
## [28,] "y"  "y"  "r"  "y"  "y" 
## [29,] "y"  "y"  "y"  "r"  "r" 
## [30,] "y"  "y"  "y"  "r"  "y" 
## [31,] "y"  "y"  "y"  "y"  "r" 
## [32,] "y"  "y"  "y"  "y"  "y"