Chapter 13.8, Problem 11PPS

To determine

The probability value of the randomly choosing password of the $3$ digits is to be estimated.

Answer to Problem 11PPS

The probability value of the randomly choosing password of the $3$ digits is $6$ .

Explanation of Solution

Given:

Conditions are:

Digits can repeat.

The password is made up of $3$ digits.

Calculation:

Consider:

Password is made up of $3$ digits $\left(r=3\right)$ as $1,2,3$ $\left(n=3\right)$

Then,

The total number of outcomes $P$ of a $3$ digits password is given by:

  $\begin{array}{l}{}^n{P}_r=\frac{n!}{\left(n-r\right)!}\\ {}^3{P}_3=\frac{3!}{\left(3-3\right)!}\\ {}^3{P}_3=\frac{3!}{0!}\left[\because 0!=1\right]\\ {}^3{P}_3{}_r=3!=3\times 2\times 1=6\\ {}^3{P}_3=6\end{array}$

Conclusion:

Thus, total number of outcomes are ${}^3{P}_3=6$ .