Chapter 12.6, Problem 47PPE

a.

To determine

To calculate: The number of permutations that are possible using each letter only once.

Answer to Problem 47PPE

The choices are $24$ .

Explanation of Solution

Given information:

Lena wants a password that uses the 4 letters of her name.

Formula used:

A permutation is an arrangement of objects in a specific order. The expression $P{}_r$ represents the number of permutations of $n$ objects arranged $r$ at a time.

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

For any positive integer $n$, the expression $n$ factorial is written as $n!$ and is the product of integers from n down to 1.

Calculation:

According to the question,

Password has 4 letters.

Without repetition of letter we get,

First time choice will be 4.

Second time choice will be 3.

And so on.

Therefore,

  $\begin{array}{l}P{}_4=\frac{4!}{\left(4-4\right)!}\\ =\frac{4!}{0!}\\ =4!\\ =4\times 3\times 2\times 1\\ =24\end{array}$

Hence, there are $24$ permutations possible for forming a password.

b.

To determine

To explain: Whether creating password based on our name is a good idea.

Answer to Problem 47PPE

No, it is not a good idea.

Explanation of Solution

Given information:

Lena wants a password that uses the 4 letters of her name.

Formula used:

For any positive integer $n$, the expression $n$ factorial is written as $n!$ and is the product of integers from n down to 1.

Calculation:

According to the question,

No, it is not a good idea because hackers could hack the account more easily .

For example the word “ LENA” as a password is not a good idea.