Question

In Exercises 5–36, express all probabilities as fractions.

Counting with Fingers How many different ways can you touch two or more fingers to each other on one hand?

Short answer

The number of ways in which two or more fingers can touch each other on one hand is equal to 26.

Step-by-step solution

Given information

In one hand, two or more fingers are touched.

Describe combination rule

The combination rule is required to count the ways of unique combinations of r units that can be derived from n units. Specifically,the order for picking the r units is not important.

The formula is shown below:

${}^n{C}_r=\frac{n!}{\left(n-r\right)!r!}$

Compute the individual counts of unique touches

The number of ways in which two or more fingers can be touched is computed using the combination rule.

To attain the total number of ways in which two or more fingers can touch, five cases are described as follows.

Case 1: Two fingers are touched

The total number of fingers = 5

The number of fingers chosen = 2

The number of ways in which two fingers can be touched is equal to

$$\begin{gathered} {}^5{C}_2=\frac{5!}{\left(5-2\right)!2!} \\ =10 \end{gathered}$$

Case 2: Three fingers are touched

The total number of fingers = 5

The number of fingers chosen = 3

The number of ways in which three fingers can be touched is equal to

$$\begin{gathered} {}^5{C}_3=\frac{5!}{\left(5-3\right)!3!} \\ =10 \end{gathered}$$

Case 3: Four fingers are touched

The total number of fingers = 5

The number of fingers chosen = 4

The number of ways in which four fingers can be touched is equal to

$$\begin{gathered} {}^5{C}_4=\frac{5!}{\left(5-4\right)!4!} \\ =5 \end{gathered}$$

Case 4: Five fingers are touched

The total number of fingers = 5

The number of fingers chosen = 5

The number of ways in which five fingers can be touched is equal to

$$\begin{gathered} {}^5{C}_5=\frac{5!}{\left(5-5\right)!5!} \\ =1 \end{gathered}$$

Compute the total counts of unique touches

The total number of different ways in which two or more fingers can be touched is the sum of counts in all five cases.

$10+10+5+1=26$

Therefore, the number of ways in which two or more fingers of one hand can be touched is equal to 26.